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Dynamical Oceanography
Lecture notes by Jüri Elken Page 51
CHAPTER 5. Shallow water theory In the shallow water the characteristic depth H
~ is much smaller than the horizontal scale of motions L
~
1~~���
LH
� .
(5.1)
It is considered that the flow does not depend on depth. This is exactly true for barotropic ( const0 �� �� ) non-frictional ( 0���� ) motions. Turbulent viscosity introduces vertical current shear (see drift currents, p. 4.3). However, vertically integrated volume transports are affected by the surface and bottom stresses only, see (4.68) and (4.69). In this chapter we consider mainly the small-amplitude wave motions. We assume that water level deviation � is much smaller than depth H
H��� (5.2) and characteristic velocity U
~ is much smaller than characteristic phase speed of the wave
TL
U ~~
~�� ,
(5.3)
where T~ is characteristic wave period. 5.1. Vertically integrated continuity equation We will introduce the following refinements as compared to the steady Sverdrup regime (p. 4.4): 1) in the non-stationary motion, vertical velocity at the undisturbed sea surface 0�z is
kinematically equal to the time derivative of the water level t�
�� (in case of large sea
level deviations we have to use the full derivative)
tw z �
���
�0
(5.4)
2) in the basin of variable depth yxH , , a free-slip bottom boundary condition (missing
velocity projection normal to the bottom) is
01,,,, ����
���
�
��
��
���� ���� yH
xH
wvuHv HzHz
�
(5.5)
Dynamical Oceanography
Lecture notes by Jüri Elken Page 52
or
���
���
�
��
���
�� ������ yH
vxH
uw HzHzHz .
(5.6)
Doing the vertical integration of the continuity equation (2.24) we obtain
���
���
�
��
���
���
��
������
����
���
����
���������
yH
vxH
ut
wwdzzw
dzyv
dzxu
HzHz
HzzHHH
�
0
000
(5.7)
Due to the small water level deviations (5.2) we may define the volume transport by integrating the velocities from the bottom to the undisturbed surface
��
�0
H
dzuU , ��
�0
H
dzvV .
(5.8)
By differentiating the volume transports, we use the rules for definite integrals with variable bounds
xH
udzxu
dzuxx
UHz
H H ��
���
���
���
��� �� �0 0
,
(5.9)
yH
vdzyv
dzvyy
VHz
H H ��
���
���
���
��� �� �0 0
.
(5.10)
Combining (5.7) with (5.9) and (5.10) we obtain the vertically integrated continuity equation
���
���
�
��
���
����
yV
xU
t�
.
(5.11)
Time derivative of water level depends on the divergence of volume transport. When the volume transport is directed towards the coast then water level will rise. Indeed, if the coast is
placed to the right from the water at Lx � , then 0�LU , 0���
LxU
and 0���
Lt�
.
5.2. Vertically integrated momentum equations By modifying the horizontal momentum equations (2.20), (2.21) we assume:
1) velocities are small (5.3) and non-linear advection terms �yu
vxu
u��
��
, may be
neglected 2) horizontal turbulent viscosity is neglected 0��
Dynamical Oceanography
Lecture notes by Jüri Elken Page 53
3) density is constant 0�� � 4) due to hydrostatic approach and constant density, pressure depends on the sea level � and
vertical coordinate
��� 000 gzgpp ��� (5.12) from where the horizontal pressure gradients depend on the sea level gradients only
xg
xp
��
��� �
�0
1 ,
yg
yp
��
��� �
�0
1 .
(5.13)
In the above assumptions we integrate the x -component momentum equation
���
���
���
����� 0
H
dzzu
zxgfv
tu
��
and similar equation for the y -component. Following (5.8) we obtain the equations for the volume transport components [m2s-2]
Hzz zu
zu
xgHfV
tU
��� ��
���
���
�����
���
0 ,
(5.14)
Hzz zv
zv
ygHfU
tV
��� ��
���
���
�����
���
0 ,
(5.15)
where viscous stresses appear at the surface
yx zv
zu
z ���� ���
���
� ,:0
(5.16) and on the bottom
ybxb zv
zu
Hz ���� ���
���
�� ,: .
(5.17) Wind stress components yx �� , are parameterized by a quadratic function of the wind velocity (2.38). Most problematic item in the shallow water models is the bottom friction parameterization since the near-bottom current shears are not explicitly resolved. In the Stommel circulation problem (p. 4.4) the linear bottom friction formulation
Dynamical Oceanography
Lecture notes by Jüri Elken Page 54
rVrU ybxb �� �� , (5.18)
could be used since the situation was anyhow idealized. It can be shown that in a quasi-steady flow regime the magnitude of the bottom stress is proportional to the magnitude of the volume transport. However, the vectors are deflected depending on the direction of wind and geostrophic flow and on the ratio of the Ekman layers to the whole water column thickness. In non-stationary wave-dominated flows quadratic bottom friction
222 VU
HUcd
xb ��� , 222 VU
HVcd
yb ��� (5.19)
gives usually better results. Here the quadratic drag coefficient is taken usually 3105.2 ���dc but it may also take into account the bottom roughness. Three-dimensional ocean circulation models based on the full set of hydro-thermodynamic equations (2.20)-(2.30), use vertically integrated momentum equations (5.14) and (5.15) in order to calculate barotropic pressure gradients based on the water level continuity equation (5.11). Bottom friction, effects of non-linear advection terms and baroclinic pressure gradients due to density variations are calculated by the three-dimensional part of the model and their effects can be considered as a correction to the wind stress. 5.3. Shallow water equations, numerical solution a) equations Vertically integrated momentum equations and continuity equation form together the shallow water equations
xbxxgHfV
tU
���
����
�����
,
(5.20)
ybyygHfU
tV
���
����
�����
,
(5.21)
���
���
�
��
���
����
yV
xU
t�
,
(5.22)
where three unknowns �,, VU can be solved using the three equations (5.20) – (5.22). In addition to the equations, boundary conditions are needed. In the case of free slip the volume transport may be directed along the coastline yx,� but normal to the coast component nU has to be zero. In the open boundary (river mouth, connection to the other sea area) normal component tyxQ w ,,� of the volume transport is prescribed
Dynamical Oceanography
Lecture notes by Jüri Elken Page 55
���
��
boundaryopentheat,,boundaryrigidtheat0
tyxQU n .
(5.23)
At the open boundary sea level or sea level gradient may be prescribed instead of normal volume transport (5.23). b) numerical solution In a realistic basin with complicated coastline and bottom topography, shallow water equations (5.20)-(5.23) are solved numerically. In a most commonly used finite difference method, temporal and spatial derivatives are written as differences along the finite time and/or grid steps. Explicit time stepping is most easy to implement. Consider evolution equation for a variable
� in a form ��f
t�
��
, where f is a spatial operator from the modeled variables.
Explicit time scheme is written as
nnn
ft
���
����1
,
(5.24)
where superscript denotes the number of subsequent time layer and t� is the value of discrete time step. This way all the spatial derivatives etc not containing the time derivative are taken from the previous time layer. For the numerical stability reasons, time discretization of the shallow water equations is quite often done as follows
nxbx
nn
nn
xgHfV
tUU
���
����
������1
,
(5.25)
n
yby
nn
nn
ygHfU
tVV
���
����
������1
,
(5.26)
���
���
�
��
��
���
�� ���
yV
xU
t
nnnn 111 �� ,
(5.27)
where in the continuity equation (5.27) the volume transports at time layer 1�n are taken from (5.25), (5.26). For the spatial discretization of (5.25)-(5.27) the domain is covered by a grid with a step
yx ��� . With a discrete set of values, the subscript i runs along the x -axis and j along the y -axis. In the Arakawa C-grid the points of definition of �,, VU are staggered as shown in Fig. 5.1. Depths are given at � - points.
Dynamical Oceanography
Lecture notes by Jüri Elken Page 56
Figure 5.1. Placement of points for volume transport and sea level on the Arakawa C-grid, used for numerical solution of the shallow water equations. Sea level derivative along x is defined at the jiU , -point
xxjiji
U ji �
��
�� � ,1.
.
���
(5.28)
and derivative along y is defined at the jiV , -point
xyjiji
V ji �
��
�� �1,.
.
��� .
(5.29)
We reach the following scheme
tHHxtg
VtfUU nxbx
nji
njijiji
nji
nji
nji �����
��
���� ��� ���� ,1,,1,,,
1, 2
~ ,
(5.30)
tHHxtg
UtfVV nyby
nji
njijiji
nji
nji
nji �����
��
���� ��� ���� 1,,1,,,,
1, 2
~ ,
(5.31)
1,
11,
1,
1,1
1 ���
���
� �����
�� nji
nji
nji
nji
nn VVUUxt
�� ,
(5.32) where due to the shift of jiU , and jiV , -points, spatially averaged volume transports have to be taken into account in the terms of Coriolis force
1,11,,1,, 41~
���� ���� jijijijiji UUUUU , (5.33)
Dynamical Oceanography
Lecture notes by Jüri Elken Page 57
1,1,11,,, 41~
���� ���� jijijijiji VVVVV .
(5.34) For the numerical stability, time and grid step must follow the relation
maxgHtx��
��
.
(5.35) which physical content is: long gravitational wave is not allowed to propagate during one time step further than one spatial grid step. In the problems of water level and coastal circulation, the numerical model (5.30)-(5.34) quite often gives more than 90% of accuracy as compared to the much more complex three-dimensional models. 5.4. Long gravity waves a) one-dimensional waves in a narrow channel Consider frictionless ( 0�� ybxb �� ) free motions 0�� yx �� in a long and narrow channel
oriented along the x -axis. Then cross-channel volume transport is missing 0�V and Coriolis force will not appear. The equations (5.20)-(5.22) obtain the form of one-dimensional waves
xgH
tU
��
���� �
,
(5.36)
xU
t ��
�����
.
(5.37) Differentiating (5.36) by t and (5.37) by x we obtain hyperbolic differential equation
02
2
2
2
��
��
�
�
xU
gHtU
.
(5.38)
In case of constant depth search for the wave solution in a form
tkxiUU ��� e0 . (5.39)
Replacing (5.39) into (5.38) we obtain that (5.38) is satisfied if the wave parameters follow
022 �� gHk� . (5.40) From the condition (5.40) we get the dispersion relation for one-dimensional long gravity waves
Dynamical Oceanography
Lecture notes by Jüri Elken Page 58
kgH�� , (5.41)
that relates frequency � to the wave number k . Ratio of frequency and wave number k
c�
�
determines the phase speed.
gHc � (5.42)
Long one-dimensional gravity waves are non-dispersive, i.e. the phase speed does not depend on the wave number(s). If we add (variable) wind stress to the equations (5.36)-(5.37) then we obtain forced waves. Here resonant forcing appears if the wind pattern moves with a phase speed of waves
gHc � . Resonant forcing is often a precondition of creating highest storm surges. Taking the mean depth of the Gulf of Finland 60 m, we obtain floods in St. Petersburg when the cyclone is moving with a speed 2.24608.9 �� m/s. Consider next the case of narrow channel of a certain length L . At the ends of the channel the volume transport has to vanish 0,,0 �� tLUtU . (5.43)
Search for the solution of (5.38) in a form
xktUU nsincos0 �� . (5.44) Boundary conditions (5.43) can be satisfied at discrete set of wave numbers
...2,1, �� nL
nkn
(5.45)
The solution (5.44), (5.45) represents the standing waves in the channel. These eigenoscillations are called seisches. Oscillations corresponding to the different discrete wave numbers are called modes (Fig. 5.2).
1
23
1
2
3
Figure 5.2. First, second and third modes of eigenoscillations of a narrow channel: volume transport (left) and water level (right).
Dynamical Oceanography
Lecture notes by Jüri Elken Page 59
b) two-dimensional waves Coriolis force may become important for two-dimensional waves. Assume constant depth
const�H . Search for the periodic solution of (5.20)-(5.22) in a form
tiyxUU �e,~� , tiyxVV �e,~
� , tiyx ��� e,~
� . (5.46)
For the spatial parts of (5.46) we obtain the equations
xgHVfUi
��
����
�~
~~ ,
(5.47)
ygHUfVi
��
����
�~
~~ ,
(5.48)
���
���
�
��
���
��yV
xU
i~~
~�� .
(5.49)
Solve for the spatial parts of volume transports from (5.47), (5.48)
���
~1~22 gH
yf
xi
fU ��
�
���
�
��
���
�� ,
(5.50)
���
~1~22 gH
yi
xf
fV ��
�
���
�
��
���
��
� .
(5.51)
Substituting (5.50), (5.51) into the continuity equation (5.49) we obtain
0~
~~ 22
2
2
2
2
��
��
��
�
��
���gH
fyx
.
(5.52)
In case of periodic wave solution �
~ has the form
lykxi �� exp
~0�� . (5.53)
Replacing this into (5.52) we obtain dispersion relation of long gravity waves on a rotating frame in the f - plane approximation
2222 lkgHf ���� . (5.54) When both the wave numbers lk , are real, i.e. the wave profile is periodic in both the x and y directions, then frequency � is higher than the inertial frequency f�� . Consequently, periodic (in both directions) gravity waves can not have a longer period than the inertial period h14�� fTT .
Dynamical Oceanography
Lecture notes by Jüri Elken Page 60
In case of longer periods 022 �� lk must take place or at least one of the wave numbers have to be imaginary. In this direction the wave profile is exponential. It is possible only near the coast whereas the wave amplitude is maximal at the coast and decays exponentially seaward. Waves with an exponential profile are called Kelvin waves. Kelvin waves appear for example at Baltic Sea seisches and also at diurnal tides. For the seisches of small basins (like the Gulf of Riga) rotational effects do not have remarkable effect. 5.5. Water level equation When investigating the wave properties apparent in the shallow water equations at different coastline and bottom geometry then it is useful to transform the equations into one equation. Frictionless ( 0���� ybxbyx ���� ) small amplitude motions are described [see (5.20)-(5.22)] as
xgHfV
tU
��
����� �
,
(5.55)
ygHfU
tV
��
����� �
,
(5.56)
���
���
�
�
��
��
����
yV
xU
t�
.
(5.57)
Let us differentiate (5.57) by time
02
2
���
��
���
��
��
�tV
ytU
xt�
.
(5.58)
Replacing tV
tU
��
��
, in (5.58) from (5.55), (5.56) we obtain
02
2
����
���
�
��
���
����
���
�
��
��
���
��
��
�yU
xV
fy
Hyx
Hx
gt
��� .
(5.59)
For eliminating VU , we take once more the time derivative from (5.59)
02
2
����
���
�
��
��
���
��
�!"
#$%
&���
���
�
��
��
���
��
��
���
tU
ytV
xf
yH
yxH
xg
tt���
.
(5.60)
Let us transform the last term of (5.60), replacing tV
tU
��
��
, from (5.55), (5.56) and taking
into account const�f
Dynamical Oceanography
Lecture notes by Jüri Elken Page 61
��
��
,2
2
HJfgt
f
xH
yyH
xgf
yV
xU
ft
Uyt
Vx
f
���
�
����
���
�
��
��
���
��
����
���
�
��
���
�����
���
�
��
��
���
��
.
(5.61)
where
ya
xb
yb
xa
baJ��
��
���
��
�,
(5.62)
is the Jacobi operator. Taking into account (5.61), the equation (5.60) is rewritten
022
2
����
���
�
��
��
���
��
�!"
#$%
&���
���
�
��
��
���
��
����
���
��
��
��
xyH
yxH
fgy
Hyx
Hx
gftt
����� ,
(5.63)
that is equivalent to the initial equations (5.55)-(5.57). Consider a boundary condition for (5.63) at wall, which is perpendicular to the x -axis. In that
case 0�U and 0���
tU
. From (5.55), (5.56) we obtain x
gHfV��
����
, y
gHtV
��
���� �
or
002
���
���
�'�
yf
txU
�� .
(5.64)
Analogically the boundary condition is obtained for the wall perpendicular to the y -axis
002
���
���
�'�
xf
tyV
�� .
(5.65)
5.6. General wave solution in a channel of constant depth With constant depth const�H the equation (5.63) is simplified
02
2
2
22
2
2
�!"
#$%
&���
���
�
��
���
����
���
��
��
��
yxgHf
tt��
� .
(5.66)
The highest order of derivatives is two and (5.66) is further reduced to the hyperbolic wave equation
02
2
2
22
2
2
����
���
�
�
��
�
����
�
���
��
�
�
yxgHf
t��
� .
(5.67)
Dynamical Oceanography
Lecture notes by Jüri Elken Page 62
Consider a channel along x -axis with a width L . Boundary conditions at the channel walls
0�V at Ly ,0� (5.68) are transformed according to (5.65) into a form
02
���
���
�x
fty
�� at Ly ,0� .
(5.69)
Search for the solutions periodic by x and t
tkxiy ��� �� e~
Re (5.70)
where y�
~ is complex wave amplitude across the channel, k is along-channel wave number
and � is wave frequency. Replacing (5.70) into the equation (5.67) and boundary conditions (5.69), we obtain an eigenvalue problem for �
~
0~
~2
22
2
2
����
���
��
�� �
��k
gHf
dyd
,
(5.71)
0~
~�� �
�� k
fdyd
at Ly ,0� ,
(5.72)
which general solution is
lyBlyA cossin~
��� , (5.73)
whereas by (5.71)
222
2 kgH
fl �
���
.
(5.74)
Here l is the cross-channel wave number ( y -component of the wave vector lkk ,�
�).
Substituting general solution (5.73) into the boundary conditions (5.72), we get a homogeneous system of linear equations with respect to BA,
0�� Bfk
lA�
,
(5.75)
0sincossincos ���
��
� ����
��
� � lLllLk
fBlLk
flLlA��
.
(5.76)
The above system has non-trivial solutions only then if the determinant is zero
Dynamical Oceanography
Lecture notes by Jüri Elken Page 63
0sin2222 ��� lLgHkf �� . (5.77) By the condition (5.77) the eigenvalue problem (5.71), (5.72) has three types of solutions, whereas the case 022 �� f� or f(�� is spurious. 5.7. Poincare waves Consider the waves corresponding to the case
0sin �lL (5.78) in the condition (5.77) of the eigenvalue problem (5.71), (5.72). Condition (5.78) is satisfied if cross-channel wave number l takes discrete values
....,2,1�� nL
nln
.
(5.79)
Seeing the definition (5.74) of l based on the equation (5.67) and wave parameters (5.70), we obtain
2
222
222
Ln
kgH
fl n
���
�� ,
(5.80)
where frequency is found as
���
���
���(�� 2
2222
Ln
kgHfn
�� ,
(5.81)
or
222nn lkgHf ��(�� . (5.82)
Dispersion relation of Poincare waves (5.82) coincides with that of the two-dimensional waves in the unbounded basin (5.54), only the cross-channel wave numbers obtain only discrete values due to the boundary conditions. Frequency of Poincare waves is always higher than the inertial frequency fn �� and the frequency of one-dimensional cross-channel
eigenoscillations nn lgH�� , or the period is shorter than the inertial period
hf
TT fn
n 1422
����
�
and the period of cross-channel eigenoscillations
gHn
LTT Ln
2�� .
Dynamical Oceanography
Lecture notes by Jüri Elken Page 64
Poincare waves can propagate both in positive and negative directions of x -axis. Phase speed
kc n
n
�� is determined from the dispersion relation (5.82) as
2
22
klgHf
gHc nn
��(� .
(5.83)
See that Poincare waves are dispersive. Remind that non-rotational gravity waves have non-dispersive phase speed gHc � . Spatial pattern of Poincare waves is derived from (5.73) where coefficients BA, in
ylBylAy nnn cossin~
��� must satisfy boundary conditions (5.72) pre-assuming discrete wave numbers (5.79). The traveling wave is
tkxL
ynfknL
Lyn
�
�
�� ���
��
� �� cossincos0
(5.84)
or
tkxyllkf
yl nn
n ��
�� ����
���
��� cossincos0 .
(5.85)
Cross-channel water level pattern of Poincare waves is asymmetric and depends on the direction of wave propagation. With 0�� the wave propagates to the right and higher water levels appear in the upper half of the channel. With 0�� the situation is opposite (Fig. 5.3).
Figure 5.3. Water level distribution in a channel due to Poincare waves propagating to the right (above) and to the left (below).
Dynamical Oceanography
Lecture notes by Jüri Elken Page 65
Superposition of two Poincare waves of equal amplitude, traveling in opposite directions, yields standing Poincare wave
tkxyln ��� coscoscos0� (5.86) where the water level distribution is symmetric and the nodal lines (lines of zero water level) are fixed. It is useful to scale the wave frequency by inertial frequency and rewrite the dispersion relation (5.81) in a non-dimensional form
!!"
#
$$%
&��
��
����
�
���
���
22
2
241
LRnR
fd
x
dn
)
� ,
(5.87)
where kx
)
2� is wavelength [m] and the barotropic (external) Rossby deformation radius
[m] is
fgH
Rd � .
(5.88)
Period of Poincare waves is determined from (5.87) as nf
n fTT
�� . At large wavelengths
*+x) the period of Poincare waves asymptotically approaches 2
2
241
1
��
��
��
�*
LRn
TTd
f
.
By increasing the channel width *+L the period approaches to the inertial period
fTT +* . At small wavelengths nL
x
2��) the wave period is proportional to the wavelength
d
x
RT
)
2� (Fig. 5.4).
Using the wavelength kx
)
2� along the x -axis and the relation (5.79)
Ln
ln
� , we obtain
the ratio of the phase speeds of the Poincare waves (5.83) and the one-dimensional waves gH in a form
!!"
#
$$%
&��
��
����
�
���
���
2
2
2
241
1LRn
RgH
c d
d
xn
)
.
(5.89)
Dynamical Oceanography
Lecture notes by Jüri Elken Page 66
Non-dimensional periods f
n
TT
and phase speeds gH
cn as the functions of non-dimensional
wave length d
x
R)
and channel width dRn
L2 are shown in Fig. 5.4. With H =100 m and
f =1.25*10-4 s-1 we obtain gH = 31.3 m/s (phase speed of one-dimensional waves) and
��fgH
Rd 250 km (barotropic Rossby deformation radius). Considering typical width of
the Baltic Sea L = 250 km, we get the non-dimensional width for the first mode dRL2
= 2 and
for the second mode dR
L22
= 1.
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0 1 2 3 4 5
Wave length )))) x /R d
Per
iod
T/T f
2L / n R d = 1
2L / n R d = 2
2L / n R d = 3
0
1
2
3
4
5
6
0 1 2 3 4 5
Wave length )))) x / R d
Pha
se s
peed
c
/ (g
H)½
2L / n R d = 3
2L / n R d = 2
2L / n R d = 1
Figure 5.4. Non-dimensional periods (left) and phase speeds (right) of Poincare waves as a function of non-dimensional along-channel wave length in case of different non-dimensional channel width. 5.8. Kelvin waves Kelvin waves are obtained from the eigenvalue problem (5.71), (5.72) considering the case
kgH(�� (5.90)
from the condition (5.77). The phase speed gHk
c (���
does not depend on the wave
number and therefore the Kelvin waves are non-dispersive. From (5.74) we obtain
gHf
l2
2 ��
(5.91)
which means that the cross-channel wave number is imaginary
Dynamical Oceanography
Lecture notes by Jüri Elken Page 67
gH
fil (�
(5.92)
and the cross-channel structure (5.73) is exponential
���
���
���
��
���
���
gH
fyB
gH
fyA expexp
~�
(5.93)
For the wave traveling in positive direction of the x -axis ( 0�� ), the boundary condition (5.72) can be satisfied only then if the water level amplitude decreases with y -axis
tkxgH
fy��� ˆcosexp0 ��
��
���
���� ,
(5.94)
where �� �ˆ . Kelvin wave traveling in the opposite direction can exist only then if the water
level amplitude is maximal at the wall Ly �
tkxgH
Lyf��� ˆcosexp0 ��
��
���
� ��� .
(5.95)
This way direction of cross-channel amplitude decay depends on the direction of wave propagation. Looking in the direction of Kelvin wave propagation, the amplitude increases to the right in the Northern Hemisphere (Fig. 5.5).
z
y
x
LH
c
Figure 5.5. Water level deviations in the Kelvin wave traveling in the direction c�
.
Dynamical Oceanography
Lecture notes by Jüri Elken Page 68
The scale for the amplitude decay in a cross-channel direction is fgH
Rd � that is also
known as the barotropic (external) Rossby deformation radius. For example at H = 100 m, the phase speed is gHc � =31.3 m/s and the e-fold amplitude decay occurs in a distance
dR =250 km. In the shallow water body with H =10 m (for example Lake of Peipsi) the phase speed of Kelvin wave is c = 10 m/s and the e-fold decay distance is dR = 80 km. Within the Kelvin wave, the cross-channel volume transport component is exactly zero
0�V (5.96) everywhere in the channel. Since the cross-channel variation of properties is monotonic then the boundary conditions (5.68) cannot be otherwise satisfied. Due to (5.96) the shallow water equations describing Kelvin waves are
xgH
tU
��
���� �
,
(5.97)
xU
t ��
�����
,
(5.98)
yfgH
U��
���
.
(5.99)
As evident from (5.99) the along-channel volume transport is in a geostrophic balance with the cross-channel sea level gradient. 5.9. Amphidromic systems of Kelvin waves Superposition of two Kelvin waves of equal amplitude propagating in opposite directions is [see (5.94) and (5.95)]
tkxR
LyRy
tkxR
LyRy
dd
dd
��
�����
ˆsinsinexpexp
ˆcoscosexpexp
0
0
!!"
#
$$%
&���
���
����
���
� ����
�
���
���
�!!"
#
$$%
&���
���
����
���
� ����
�
���
����� ��
,
(5.100)
where dR is the deformation radius (5.88). Expression (5.100) represents the standing Kelvin wave which general form is
��
��
� ������2
,ˆsin,,ˆsin,ˆcos, 22
2121
,���� yxtyxAyxAtyxAtyxA ,
(5.101)
Dynamical Oceanography
Lecture notes by Jüri Elken Page 69
where the initial phase dependent on yx, is
�
��
���
��
yxAyxA
yx,,
arctan,1
2, .
(5.102)
Expression (5.101) means that the amplitude of oscillations at every point is given by
yxAyxAyxA ,,, 22
21 �� (5.103)
and the co-range lines are determined by const, �yxA . In the case when the amplitude (5.103) is permanently zero 0, �yxA at some isolated points, the waves form amphidromic systems and the points of zero amplitude are nodal points of the amphidromic system. Note that initial phase (5.102) cannot be determined in the nodal points. Outside the nodal points the phase of oscillations is different depending on the initial phase (5.102). Co-tidal lines are determined by the equation
yxt ,ˆ ,� � or yxA
yxAt
,,
ˆtan1
2�� .
(5.104)
Co-tidal lines converge in the nodal points. The lines are usually marked by angle 00-3600 or
time � 2
0 -�t .
Amphidromic systems may appear at different waves. In the oceanography they are mainly related to the Kelvin waves arising from the tidal forces and from the excitation of seisches by the changing winds. In view of (5.100) and (5.101), standing Kelvin waves have specific amplitude distribution in the channel
kxR
Ly
RL
Add
cos2cosh2
exp01
����
�
����
�
� �
���
���
��� � , kx
R
Ly
RL
Add
sin2sinh2
exp02
����
�
����
�
� �
���
���
��� � ,
where nodal points are 20
Ly � ,
kn
x
�0 . Using the expansion into Taylor series
dd Ryy
Ryy 00sinh
����
�
���
� � and 1cosh 0 ���
�
���
� �
dRyy
if 10 ���
dRyy
, the equation for co-tidal lines
(5.104) is transformed as
kxR
yyt
d
tanˆtan 0��� .
(5.105)
Since kxkx �tan at 1��kx , the co-tidal lines become strait lines near the nodal point
0ˆtan ytxkRy d �� � , (5.106)
Dynamical Oceanography
Lecture notes by Jüri Elken Page 70
where tangent of the line tkRd �̂tan increases with time t . Therefore in the Kelvin waves, the oscillation phase rotates in the Northern Hemisphere counterclockwise around the nodal points (Fig. 5.6a). Amphidromic systems of Kelvin waves appear also in the closed and semi-enclosed basins (Fig. 5.6b). a)
b)
Figure 5.6. Amphidromic systems: a) Kelvin waves in an infinitely long channel, b) M2 tides in the North Sea. 5.10. Topographic Rossby waves Consider a channel with sloping bottom
��
��
���
Ly
HH.
10 ,
(5.107)
where the bottom slope 1��. is small. We can take 0HH � in the equation of water level
(5.63), except for the derivatives where we obtain L
HyH .
0����
. The water level equation
takes the form
0002
2
2
2
02
2
2
���
�!"
#$%
&
��
����
���
�
��
���
����
���
��
��
��
xLfgH
yLgH
yxgHf
tt�.�.��
� .
(5.108)
Dynamical Oceanography
Lecture notes by Jüri Elken Page 71
Search for the wave solution as earlier (5.70)
tkxiy ��� �� e~
Re . (5.109)
The cross-channel structure y�
~ must satisfy
0~
~~~~ 00
2
2
02
022 ��!
"
#$%
&������ �
.�.�����
LkfgH
idyd
LgH
dyd
gHkgHfi
(5.110)
or after simplifications
0~
~~2
0
22
2
2
����
���
���
��� �
�.��.�L
kfk
gHf
dyd
Ldyd
.
(5.111)
Boundary conditions are the same as earlier (5.72)
Lykf
dyd
,0,0~
~��� �
��
.
(5.112)
General solution of the equation (5.111) has the form
lyBlyALy
cossine~ 2 ��
.
� ,
(5.113) where
2
22
0
222
4LLfk
kgH
fl
.�
.����
�� .
(5.114)
Boundary conditions (5.112) yield the eigenvalue problem 0sin2
0222 ��� lLkgHf �� , (5.115)
that is similar to (5.77) in the case of channel of a constant depth. We will see that Poincare and Kelvin waves remain nearly unchanged at small bottom slopes,
only a slowly changing multiplier Ly
2e.
appears in the wave amplitude. From the condition
0sin �lL (5.116) of the problem (5.115) we obtain discrete cross-channel modes
Dynamical Oceanography
Lecture notes by Jüri Elken Page 72
Ln
ln
� .
(5.117)
From (5.114) we obtain cubic equation for the frequency
04 2
2
0
2
2
222
002 ���
�
���
������
LgHf
Ln
kgHLfkgH . �
.� .
(5.118)
In case of low-frequency waves
�.
�LfkgH 02 �� ,
(5.119)
using also condition for small bottom slope 0
2
2
222
2
2
4 gHf
Ln
kL
���� .
, we obtain the
dispersion relation for the (barotropic) topographic Rossby waves
0
2
2
222
gHf
Ln
k
kLf
��
��
.� ...,2,1�n
(5.120)
Frequency of topographic wave is maximal (period is minimal) at the scales of Rossby deformation radius
0
2
2
22
gHf
Ln
kk n ���
or 22
22 1
d
nRL
nk ��
(5.121)
Topographic Rossby waves can propagate in one direction only: the shallower water must remain to the right from the direction of wave propagation. Topographic waves are quite slow and the phase speeds are usually less than 1 m/s. For the small bottom slopes, the frequency of topographic waves is always less than the inertial frequency f . All the barotropic waves considered above in p. 5.7-5.10 are summarized in Fig. 5.7 adopted from J. Pedlosky (“Geophysical Fluid Dynamics”, 1979).
Dynamical Oceanography
Lecture notes by Jüri Elken Page 73
Figure 5.7. A scheme of dispersion relations of the barotropic waves appearing in the in finitely long channel: non-modal Kelvin waves propagating in both directions, cross-channel modes of Poincare waves propagating in both directions, cross-channel modes of topographic Rossby waves propagating with shallower water on the right (a figure by J. Pedlosky).