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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 ~ ~ L H . (5.1) It is considered that the flow does not depend on depth. This is exactly true for barotropic ( const 0 ) 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 T L 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) t w z 0 (5.4) 2) in the basin of variable depth y x H , , a free-slip bottom boundary condition (missing velocity projection normal to the bottom) is 0 1 , , , , y H x H w v u H v H z H z (5.5)

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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).