MATEC Web of Conferences 25 , 0 1 0 1 4 (2015)
DOI: 10.1051/ m atec conf/ 201 5 2 5 0 1 0 1 4
C Owned by the authors, published by EDP Sciences, 2015
Influence of Complete Coriolis Force on the Dispersion Relation of
Ocean Internal-wave in a Background Currents Field
Yongjun Liu*
College of Economic and Management, Linyi University, Linyi, Shandong, China
Xiaoping Gao & Tianxia Yu
Department of Mechanical and Electrical Engineering, Shandong Water Polytechnic, Rizhao, Shandong, China
Jun Yang
College of Economic and Management, Linyi University, Linyi, Shandong, China
ABSTRACT: In this thesis, the influence of complete Coriolis force (the model includes both the vertical and
horizontal components of Coriolis force) on the dispersion relation of ocean internal-wave under background
currents field are studied, it is important to the study of ocean internal waves in density-stratified ocean. We start
from the control equation of sea water movement in the background of the non-traditional approximation, and the
vertical velocity solution is derived where buoyancy frequency N(z) gradually varies with the ocean depth z. The
results show that the influence of complete Coriolis force on the dispersion relation of ocean internal-wave under
background currents field is obvious, and these results provide strong evidence for the understanding of dynamic
process of density stratified ocean internal waves.
Keywords:
Coriolis parameter; internal waves; dispersion relation; non-traditional approximation.
1 INTRODUCTION
A number of observational data show both the vertical
and horizontal of the ocean internal-wave which is
especially not negligible in this study of near inertial
internal waves (Ahran [1], Boebel and Zenk [2] and so
on), the parameterization problem of the parameters
and explain the dispersion relation of ocean internal
wave, wave remote sensing information extraction and
ocean model of the internal wave. Considering the
horizontal component of non-traditional approximation, the recent research for propagation characteristics
of linear internal waves in the ocean have aroused
wide concern, and many authors such as Kundu and
Thomson [3], Gill [4], Kasahara and Gary [5], Gerkema
and Shrira [6], Garrett and Munk [7] and so on have
studied the internal waves in different forms of approximate solution and numerical solution, but these
studies did not give a simple and practical internal
waves’ dispersion relation expressions without the
effect of background flow field on the frequency dispersion relation under non-traditional approximation.
Therefore, we provide a normal mode analysis of the
linearized Boussinesq model with particular emphasis
on internal waves’ dispersion relation with the approximate analytical solutions, and the dispersion
relation was compared between the non-traditional
and traditional approximation in a background flow
field. The results show that the influence of complete
Coriolis force on the ocean internal wave dispersion
relation in a background currents field is not negligible.
2 DISPERSION RELATION OF A BOUSSINESQ
APPROXIMATION WITH ROTATION
2.1 Basic Equations
It is assumed that the fluid is irrotational and incompressible for continuous density stratification of seawater, Boussinesq and the complete plane approximation (which is also known as a non-traditional approximation) in the linear internal waves of the governing equations (Gerkema and Shrira [6] and reference
Phillips [7]) and the horizontal uniform steady currents
field U U 0 ( z),V0 ( z ),0
Figure 1. Internal-wave motion diagram in a background
current field
*Corresponding author: lyj00001@163.com
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Article available at http://www.matec-conferences.org or http://dx.doi.org/10.1051/matecconf/20152501014
MATEC Web of Conferences
U
~
u
u
u
p
w 0 U0
V0
f v V0 fw (1)
t
z
x
y
x
V
v
v
v
p
w 0 U0
V0
f (u U 0 ) t
z
x
y
y
(2)
Then the derivative of Equation (3) is with respect
to the variable x , so:
2w
2w
2 w ~ u
2 p b
f
U 0 2 V0
(9)
xy x
xz x
x
tx
Subtract Equation (8) from Equation (9), we obtain:
w
w
w
p
U
V
f u U b
t
x
y
z
(3)
b
b
b
U0
V0
N 2w 0
t
x
y
w u w u U
t x
z x x
z (4)
V
(5)
V u U w
z y z z u v w
0
x y z
Here, the coordinates are selected as shown in Figure 1.
, p ,u , v
and
w
are respectively the
f
w u U u
y x
z z x
(10)
v V f wz ux bx
z
corresponding fluid density, the perturbation pressure
and the velocity components for (x,, y ,z );
b repre-
sents the buoyancy; f 2 sin and fˆ 2 cos are
respectively corresponding to the vertical and horizontal components of Coriolis force; is the earth's
rotation speed; represents the latitude; t is the time;
h1 and h2 are only corresponding to the different
Derive Equation (10) with respect to the variable x ,
then:
w
u w u U t x
x z z x x
V
w u U u
x y x
z z x
depth of fluids; under the f -plane approximation,
f and fˆ can be considered as constant and we de-
V u z x y x z
note
f
d
u U v V w as the
dt
t
x
y
z
derivative operator.
Rigid lid approximation is used as the upper surface
and the bottom boundary condition of the fluid motion,
w 0, z 0 ,
w 0 , z H .
(6)
(7)
Equations (1)--(5) and boundary conditions (6)--(7)
constitute the basic equations for the motion of the sea
water in a horizontal uniform current field.
At first, the derivative of Equation (1) is with respect
to the variable z, then:
u
U u
V u
u
U
t
z
z
x
x
z
z y
And then the derivative operator is used in Equation
(11), which can be:
w
u w
u u U x z t x x
x z t x
V
U v V u
w U w f y z
z z
z z z
f
w
p
w
x z
z
(8)
v V t y xw u w
u w
x z t z x
x z w u w u U u U U z z t x x
x x
v V U
V
w u w u wU z z x z x
x y x
w u w u u U V t x y x
z z x y x
v V V
(11)
w
u b
v V f x z
x x
x z
2.2 Vertical structure equations
U w
z w u w u wV
z x y z x
z x y x
V u U u
U u
u U t z x
z x y x z x V u V u U u
v V z x y y z x z x y w
V u U u
U w
z z x
z x y t x z z 01014-p.2
EMME 2015
u U x xz w
w
tt
v
v
fw
x y z
x z w
u w
u f u U x
z
x
x
z
x x
f v V w
v w
u U y z t x y t y
U v
v
f u U w
f
z t x z
x z x z
f v V f
Equation (16), which can be:
U U v V w
z x y z z The derivation process is similar to (8)--(12), and
we derive Equation (2) with respect to the variable z ,
then:
(13)
Uw
x y
w
v y z y
w
v w
V
x z y
y z t y y v w
v u U V
y z x y y y z w
v y z y y v V V
w
v V w
y z y
y z t y z z u U V w
x y z z v V y z w
(14)
p
b
y z
y
V V w
w
z y z z U v V v t y z x
z y Subtract Equation (13) from Equation (14), and
then we obtain:
w v w v w v U
V
t y
z x y
z y y
z (15)
U V U v V v u
w
f
z z z x
z y z
z
u
b
f
y
y
Then derivative of Equation (15) is with respect to
the variable y , so:
v V U v V v z y y z x
w
U v V v y z z x
z y f
u
u
f u U t y z
x y z
f v V u
b
f
y y And then the linear derivative operator is used in
u
u
fw
y z
y z u
u
f u U x y t y
f
(16)
U v V v z y z x
u U xy f v V w
v w
v U
t y
y z x y
y z U v V v u
f
y z x
z y y z
v V U V w
Then the derivative of Equation (3) is with respect
to the variable y , so:
w
v V V
w
y y
y z y z z w
v w
U
t z y
y z t x y U p
z
y z
w
w
w u
U
V
f
y ,
t
y
x
y
y
w
v w
v u U U y z y z x y
b
b
b
b
u U v V w t x
x
x y
x z
V
V
U v
w z
z x
z
V v
v
u
V
f
z y
y z
z
v w
v v V y z t y y
y z (12)
w
u w
u fw
f
y x z
z x z
x x v
w
t z
z
v
U
x z
u
u
ffw y
y z
b
b
u U t y
x y v V 01014-p.3
b
b
w y
y z
(17)
MATEC Web of Conferences
Add Equation (12) and Equation (17), and put the
result in order, then:
w w
u
v v V w u U x
y
z t x z x
y y t
w w
u
v U u U v V w x
y
z x x z x
y y t
w w
u
v V u U v V w x
y
z y x z x
y y t
V u
U v u U v V w x
y
z z x
z y x
y t
U w V w u U v V w x
y
z z z x
z y t
u
u f u U v V w x
y
z x
y t
w
f u U v V w x
y
z x z
t
b
b u U v V w x
y
z x
y t
(21)
u
v w
f f
y y
x
Take the Continuity Equation (5) into Equation (21),
so:
u v y y
x (22)
w
w
w
f
f
y
z
y
Then derive Equation (5):
(23)
With the derivative operator:
D
U
V
t
x
y
Dt
D w
w
w
Dt z x
y
U V u
v z y x
y z x
u v u v V
x y
x y y
x u
v
w
x
y
z
Which is:
d
dt
U
V
U
x
t
V w U z x
z
u
v f u U v V w x
y
z z y
x t
u v V u v t y
x z y
x V w U w
z z x
z y (24)
We obtain:
(18)
u
u
w f x z y
x
D u
v w
w
w ,(25)
U
V
Dt x
y t z
x z
y z
The derivative of Equation (22) is with respect to
the variable y , then:
V w
U w
D u
v Dt y
x y z x y
z y u
v b d b
f
z y
x dt x
y f
(26)
w
w
f
y z
y Derive Equation (1) with respect to the
variable y , then:
And the derivative of Equation (25) is with respect
to the variable x , so:
U w
u
u
u
U
V t y
z y
x y
y
D u
v w
w
w (27)
U V
Dt x
xy t xz
xy z
x z
(19)
v
w
p
w
f
f
y
x y
y
Then the derivative of Equation (2) is with respect
to the variable x , so:
V w
v
v
v
U V
t x
z x
x y
x
(20)
u
p
f
x
x y
Subtract Equation (19) from Equation (20), and
then we obtain:
Let Equation (27) add Equation (26), then we can
get:
D
Dt
u u V w
U w
z x y
z y y x
f
w
w
w
w
w
f
U V
t x z
y z
x y z
y
x z
(28)
From Equation (22), we can get:
2
3b
3b
3b
2 w
U
V
N
0
0
0
tx 2
x 3
x 2 y
x 2
(29)
And:
3b
3b
3b
2w
U0
V0 3 N 2
0
2
2
ty
xy
y
y 2
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(30)
EMME 2015
Then:
w D b b w
(31)
N Dt x
y y x
Take Equations (22), (23), (28) and (31) into Equation (18), the vertical velocity equation of the fluid
movement is as follows:
w
w d D w
dt Dt x
y z V w D U z y z Dt z x
D
Dt
U w
V w z y z z x
V
U w
V w
w
w
f
y z y z xy
f
w
w
w
f
y z
t xz
y v
v y x
D w V w
z x
Dt y z U w
w
w
f
f
z x y
x z
x y
(34)
d
w dt
(35)
2.3 Wave solution and dispersion relation
We assume that the vertical velocity has the form:
V w U w
w
f f
z y
z
z z x
f
D
Dt
wx, y, z, t W z exp i(kx ly t )
Where, k and l are the wavenumbers in x and y. By
substituting (36) into (32) and eliminating the exponential factor, we can obtain the following equations
for W(z):
f U k V l W z ilffW
ffW z w
w D w U V
f
xy z D x z .(32)
Dt
x z
N
w
w N x
y
The Equation (32) is the vertical velocity equation
of the sea water in a horizontal uniform background
current field under non-traditional approximation. If
the average background flow does not exist, then the
Equation (32) will be degenerated to the vertical velocity equation of internal-wave. Equation (32) and
boundary conditions (6) and (7) can determine the
vertical velocity of fluid motion in a horizontal uniform steady current field, and other wave field elements can be expressed by the vertical velocity w as
follows:
U
V
k z z
u
u y x
V w
U w
,
z x y
z y D
Dt
f
w
w
w
f
y z
t x z
y w
w
U V
x y z
x z
And:
(36)
U k Vl k l f l (37)
l U k Vl W z Equation (37) and boundary conditions (6)--(7)
formed the vertical velocity equation of fluid motion
in a horizontal uniform current field.
Let:
W z " z Hi! z
(38)
Where, γ is a real coefficient that will be determined
later. By substituting (38) into (23) and (37), we can
obtain the following equation:
f U k V l " z ! f
ff " z i! f U k Vl ilff
(33)
U k Vl ! lffff
N U k Vl k l f l U V
k z z
l U k Vl " z Let the coefficient of "
z
01014-p.5
equals 0, then:
(39)
MATEC Web of Conferences
! lff
ff
(40)
f U k Vl The Equation (39) is changed into the following
form:
" z
N U k V l k l f l f U k Vl U
V
k z
z j $ U k Vl H k l
U
V
k
l
z k l z k l j $ U k Vl H k l
(41)
U
V
k
l
U k Vl z k l z k l f l f j $ N f k l
H k l
ff
lff
" z U k Vl f
U k Vl l
f U k Vl j $ H k l
j $ U k Vl H k l
When not considering the effect of the background
current field, (41) reduced to the equations is the
Boussinesq version of the similar equations (7 on page
82 of Liu Yongjun [11].
Our task now is to solve (41) under the boundary
conditions (6) and (7) then
U
V
k
l U k Vl z k l
z
k
l
f l f j $ N f k l
H k l
2.4 Solutions of the system in the case of constant N
Because the boundary conditions are w= 0 at z= 0 and
1, the solution of (41) is chosen in the form. The general solutions of the system of Equation (41) can be
expressed by:
(42)
Where, κ is the vertical half-wavenumber and it is
expressed by:
# N U k V l k l f l f U k Vl (43)
f
U k Vl l
f U k Vl j$
H
U k Vl j $ U k Vl H k l
U
V
k
l U k Vl z k l z k l " z VLQ # z
lffff U k Vl U
V
k
l f z k l l z
k
U
V
k z z
And j= 1, 2, 3, … respectively denotes the vertical
wave mode index. Solution (42) as wave motion satisfies (41), and it provides that:
f l f j $ N f k l
H k l
U k Vl U
V
k
l f U k Vl z k l z k l j $ f N H k l
(44)
f This is a quartic equation in ω. Therefore, there are
four kinds of wave oscillations which propagate in
four directions with the different phase speed. The
properties of these wave modes are discussed by
Kasahara (2003a, 2004), Durran and Bretherton (2004)
for the cases of N = constant. Therefore, the readers
will refer to those references. However, we should
bring out some basics of the dispersion equation (44)
that are pertinent to understand the contrast of eigenfunctions between the four different kinds of normal
modes.
01014-p.6
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3 CONCLUSIONS
REFERENCES
In this paper, the ocean internal waves dispersion relation under non-traditional approximation derived by
Liu Yongjun et al. (2009) is extended to the situation,
which the influence of complete Coriolis force and a
horizontal uniform background current field on ocean
internal waves solution and dispersion relation is considered with the rigid lid approximate boundary conditions of the surface and bottom permeability boundary condition. And we assume that the buoyancy frequency N is constant under the f-plane approximation,
the internal wave solutions and the dispersion relation
are obtained from the linear dynamic equations. Our
study shows that there are four kinds of wave oscillations with different phase velocity which propagate in
corresponding four directions; it is different from the
conclusion of Kasahara and Gary (2006), who confirmed that there are two kinds of wave oscillations
which propagate in both eastwards and westwards
directions with the same phase speed. Only the effect
of complete Coriolis force is considered, but the influence of background current field is not considered.
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[6] Gerkema, T. & V. I. Shrira. 2005. Near-inertial waves in
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[7] Garrett, C. and W. 1979. Munk. Internal waves in the
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[8] K. Aagaard, A. T. Roach, & J. D. Schumacher. 1985. On
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ACKNOWLEDGEMENT
The authors (Liu Yongjun et al.) thank Wang Xin for
his interest and advice during the course of this research, and thank Kasahara and Gary for their information on the near-inertial oscillations in the oceans.
The authors also thank two reviewers' useful comments to increase the readability of this paper. This
manuscript was typed by Gao Xiaoping at SWP.
01014-p.7