Surface Waves

Waves and Tides

5.4 Surface Waves

(a)

-250

E . --750

- 1000

-1250

Salinity, s(psu1

3 2 33 34 35 3,6

Temperature, T ( C I

-\

- 1 5 o o - / L , , I , ,

21 22 23 24 25 26

4

27 -1600- 0 5

1 Density anomaly, mt ( k g m - 3 ) Brunt-Vaisala frequency, N i 2 r (cycles h - ' Fig. 5.2 (a) Vertical profiles of salinity, temperature, and density anomaly for a trop- ical, landlocked sea under low wind conditions. (b) Profile of buoyancy frequency calculated from data of (a). Density is controlled mainly by temperature, which is characteristic of most nonpolar waters. [Adapted from Apel, J. R., eta/., J. Phys. Ocean- ography (1985).]

nearly isohaline conditions existing at depths below approximately 200 m.

The maximum Brunt-VBisBIB frequency occurs where the temperature gradient is largest (near 150 m depth) and has a value in excess of 10 cycles h - ' (0.01745 rad s - I ) . Another example, which represents more typical temper- ate Atlantic conditions, is shown in Fig. 5.3. Here Nshows a secondary maximum near a depth of 750 dbar, probably due to the presence of a differing water mass. The calculated quantity dtl/dp on the graph is the potential temperature gradient, and its constancy beneath some 2000 dbar, despite decreasing N, indicates that the buoyancy frequency variations below that depth are due to adiabatic compressibility alone (i.e., g/c*).

Brunt -Vaisala frequency, N/2: (cycles h-')

o I 2 3 4 5 6 7 a 9 10

4000/$

4 2 5 0 g

4750

450011

5000-

-0.10 -0.05 0 0.05 0.10 0.15 0.20 0.25 0.30 Potential temperature gradient, d8idp ( 10-1 " C

0.35 0.40 dbar-1)

Fig. 5.3 Profiles of buoyancy frequency and potential temperature gradient for tem- perate Atlantic Ocean. Secondary maximum in quantities is probably due to differing water masses. [Adapted from Gill, A. E., Atmosphere-Ocean Dynamics (1982). Origi- nally from R . C. Millard.]

has been learned of the linear properties of gravity and capillary waves, and some theoretical advances have been made in understanding their weakly nonlinear features. However, it is fair to say that in spite of much attention, not only are strongly nonlinear features such as cusping and breaking not well modeled, but the very processes by which the wind-by far the most ubiquitous source of wave energy-actually generates surface waves are not altogether clear. In another arena, an extremely long wavelength type of gravity wave, the tsunami or seismic sea wave, occurs with sensible amplitude so rarely that the available observational data base is insufficient to test the- ories of tsunami generation and propagation. The deep sea tide, another well- known kind of oceanic motion, is a forced, shallow-water, linear wave that has recently yielded in some degree to the computational power of numeri- cal analysis, and reasonable scientific predictive capability now exists for open ocean tides, based on theoretical principles and observational data.

This is not to underestimate the predictive skill in surface waves and tsunami accumulated by oceanographers; indeed, operational forecasts and warnings for both types have been implemented, and produce acceptable alerts in normal situations. However, cases of severe storm waves are still misforecast with enough frequency to dilute the confidence of those who must rely upon such predictions. In addition to a lack of timely data, it is fair t o say that there are still first-order unsolved research problems in nonlinear surface waves, and their study remains an area of active interest.

The wind is thought to generate surface waves through a sequence of processes that assume varying proportions with time. At very low wind speeds, small capillary waves develop on the surface, and are thought t o be a time- and space-growing viscous instability. As the instability causes the capillaries to grow, the wavelengths and amplitudes increase so as to extend them into the gravity range (of order centimeters), whereupon new processes appar- ently take over to continue the growth. The first is direct forcing via wind pressure exerted on the steep parts of the wave slope; the second (and probably the less important) is shear stress exerted on the surface by (1) turbulent eddies in the wind, and (2) tangential surface stress from air flow over the small-scale, irregular wavelets on the surface. The pressure work done by the wind derives largely from those wind Fourier components whose wavelengths are near the water wavelengths and which are in phase with the wave slope. Figure 5.4 schematically illustrates these processes along the length of one gravity wave, and gives an indication of the streamlines of air flow in a coordinate system moving with the wave phase speed, cp.

As the waves continue to increase in length and height, yet another process comes into play: nonlinear, finite amplitude wave/wave interactions, in which the quasi-random gravity waves, in scattering off one another, produce both longer and shorter waves. The scattering process can be thought

Waves and Tides 173

Fig 5.4 Schematic of weakly breaking surface waves showing the normal pressure force near the crest of the wave and the tangential wind-stress force. The streamlines of air flow over the crests show regions of closed circulation in a coordinate system traveling at the wave phase speed. [Adapted from Banner, M. L., and W. K. Melville, d . Fluid Mech. (1976).]

of as being caused by the interactions of four waves, two of which intersect to form an interference pattern that is considered to be a third, virtual wave.

This virtual wave then scatters a fourth, real wave toward longer wavelengths.

While weak, this process is thought to be largely responsible for the characteristic spectral shape and increasing wavelengths of water waves as time goes on. The scattering toward longer wavelengths results in a continual length- ening of the dominant waves as time progresses, while scattering toward shorter lengths generates waves that are lost among the spectral components already existing at small wavelengths.

On the short wavelength side, the wave energy also spreads itself among the various frequencies (or wavelengths) present, through long-wave/short- wave interactions, so that in near-equilibrium, the wave energy distribution reaches a saturation form that is more or less global in its level and func- tional form, somewhat as the molecules in a gas reach a Maxwellian distribution when in equilibrium. Examples of this spectrum will be given ahead.

In addition to the wind speed and direction, the generation of waves clearly depends on the length of time that the wind has been blowing, i.e., the duration, and the distance that the observation point is located offshore, that is, thefetch. Both the duration and fetch that are required to allow a “fully developed sea” to be formed depend on wind velocity (Fig. 5 . 5 ) . For example, in order for a wind of 10 m s ^~

’

to generate a fully developed, saturated wave spectrum, it must blow for nearly 18 h over an oceanic expanse of perhaps 320 km. Under those conditions, the so-called significant wave height, will be approximately 2 m, and the wave period, T,, will be about 7.5 s.

However, there is an entire spectrum of waves that actually make up the “significant” wave. Figure 5.5 also provides estimates of typical wave periods, lengths, and heights. While waves of the height of this example may also be generated by stronger winds under conditions of shorter duration and less fetch, they will not have equilibrated into the global spectrum illustrated in Fig. 5.6.

Wind duration, tW (h)

Fig. 5.5 Cumulative sea state diagram showing significant wave height, as a function of wind duration, fetch, and speed. A fully developed sea (FDS) is considered as having arisen from conditions shown along the near vertical line labeled “FDS period, T,,,,,.” [Adapted from Van Dorn, W. G., Oceanography and Seamanship (1974).]

This figure shows the distribution of squared wave heights, or wave energy, among various frequencies under fully developed equilibrium conditions.

As the waves propagate away from the immediate region of generation, or as the wind speed is reduced, they become swell, which is actually the far- radiation field of the surface wave source. Since longer waves travel faster than shorter ones (i.e., they are dispersive; see Section 5.6), the wavelengths and periods of swell gradually increase with the time and distance from the source; their amplitudes are also reduced due to spreading and friction, so that swell waves are usually linear, coherent, small-amplitude gravity waves generally having periods longer than several seconds.

Having briefly described how wind waves are generated and propagate, we will now deduce some of their linear properties from the hydrodynamic equations. The surface wave problem has many characteristics that allow the exercise of the theory developed to date, but does not necessitate most of the thermodynamics. In addition, real wind waves are almost always nonlinear, but we will not consider that important complication here.

Waves and Tides 175

10-E

0.2 1 .o 10

Frequency, w (rad s-’ 1

Fig. 5.6 Height spectrum of surface gravity waves as a function of frequency, for the “equilibrium range” beyond the spectral peak; the shape of the peak is shown for only three cases. Slope of the solid straight line i s - 5 (cf. Eq. 5.109). However, more recent work suggests that the slope of the high-frequency region may actually be - 4 if analyzed differently (dashed lines). [Adapted from Phillips, 0. M., The Dy- namics of the Upper Ocean (1977).]

5.5 Linear Capillary and Gravity Waves

Surface waves have frequencies, w , much greater than N o r f, and t o a first approximation may be considered as linear, so that u . ^{V u}= 0. In addition to making the linear assumption, we may neglect the Coriolis, buoyancy, and viscous terms in the momentum equation, although dissipa- tion becomes increasingly important at higher frequencies. Under these con-

ditions, it can be demonstrated (and we will d o so after deriving expressions for the velocity) that the fluid motion will be irrotational, or will possess no vorticity. Fluid vorticity, {, is defined as the curl of the velocity field:

{ = v x u . (5.32) A vector field that is irrotational can be derived from a scalar potential, which for the fluid velocity is named appropriately enough the velocity potential,

cp (not to be confused with the potential energy per unit mass, 9, or the coordinate, 4). Thus we may write for the velocity,

u =

vcp.

(5.33)

Additionally, the incompressibility condition, V . u = 0, implies that the velocity potential satisfies Laplace’s equation,

V2p = 0 . (5.34)

A large body of mathematical methods is available to arrive at solutions to Eq. 5.34.

Consider now a uniform, frictionless ocean of depth H , whose surface is subject to a small wavelike perturbation, E(x,y,t), as shown in Fig. 5.7. In the Boussinesq approximation, the total pressure in the fluid, p , is the hydrostatic pressure, po = p a - pgz, plus the perturbation pressure, p ’ = Pgt :

5 _a

- h Fig. 5.7

P = Pa

+

p g ( - z

+ 5 )

^* ^(5.35)

cg and cP-

Horizontal distance, ( x 2 + y 2 ) %

*

Schematic showing nomenclature for surface waves.

Waves and Tides 177

Under the conditions assumed above, the equations of motion reduce to

a U

p - = - V p ‘ and

v . u = 0 ,

(5.36)

(5.37) since the perturbation density, p ’ , equals 0, by assumption. Operating on Eq. 5.36 with the divergence operator,

v

^-,one immediately obtains Laplace’s equation for the perturbation pressure, p ‘ :

a

- p - at ( V S U ) = V . V P ‘ = v 2 p ‘ = 0 . (5.38) Thus the perturbation pressure is a type of velocity potential in this instance.

Because the problem is linear, we may assume a traveling sinusoidal wave as a solution, since any arbitrary linear disturbance may be composed out of such basis functions via Fourier analysis. Now p ’ is proportional to

4 ,

so we may try as a solution

where the horizontal wave vector, k,,, is

kh = k f

+ $,

(5.40)

and the radian frequency is w . The wave moves with a phase speed, cp, given by

C p = W/kh , (5.41)

where the scalar horizontal wave number is

kh = ( k 2

+

^1’)”

.

(5.42)

We have not yet specified a

z

dependence, for this must come out of the analysis; nor have we applied the boundary conditions. Laplace’s equation (using Eqs. 5.35 and 5.39) becomes

(5.43)

The boundary conditions are those given by Eqs. 3.1 17a, 3.121, and 3.123, transformed to conditions o n p ' . Thus at the surface,

z

⁼

t ,

and at the bottom,

z

⁼- H , we impose the requirements:

W ( Z ) = a.gat a t z = t , (5.44a)

P ( Z ) = P O ( Z )

+

P ' ( Z ) = P o a t z =

t ,

(5.44b) at

z

⁼- H , (5.44~) w ( z ) = 0

and

a p w / a z =

o

z

= - H

.

(5.44d) The free surface boundary conditions can be approximately satisfied by ap- plying them at z = 0 rather than

z

⁼

t ,

^becauseof the small amplitude assumption. Hence Eqs. 5.44a and 5.44b will be evaluated at

z

= 0 when imposing the boundary conditions.

Now the time variation of all the time-dependent quantities must be harmonic, sincep

- t -

^cos^{w t .}The harmonic solution that satisfies the bottom boundary condition is

and

W = Wo sinh kh ( Z

+

^{H )}^sin( k h . X - w t )

.

^(5.46)

The horizontal velocity components, similarly obtained from Eq. 5.36, are:

It is readily established that these solutions meet the surface and bottom boundary conditions. One additional condition must be met, however, if we are to incorporate capillary waves-continuity of pressure across the inter- face, from which Eq. 3.123 was obtained:

P = P a - .,(sT a2t

+ ay2

^{a 2 t )} ^(5.49)

Waves and Tides 179

Here ^7,is the surface tension and pa the atmospheric pressure. T o proceed further, we need one of the several forms of the Bernoulli equation t o de- scribe the surface dynamics. By writing the velocity in terms of ^{V p ,}our ver- sion of the momentum equation can be rearranged to give:

v ($ ⁺

^!h^u2

⁺ ”

^-^P

⁺

^gz

> ⁼ ^vx

⁼^{0 ,} ^(5.50)

where

x

is simply an abbreviation for the quantity in the parentheses. In- tegrating this along a streamline, $, just below the free surface, along which p and p are constant, we obtain

j, ^vx-ds

⁼

^x

⁼^const.

This means that the sum

a(P PI P C

+

^‘/2^u2

+

^g[ ⁼^-⁼^const.^,

a t P P

(5.51)

( 5 . 5 2 )

is a constant, p,/p, along the streamline. Neglecting ‘/z u2 as being of second order, we may combine Eqs. 5.44a, 5.49, and 5.52 and differentiate the result with respect to time to get

(5.53)

Dalam dokumen Buku Principles of Ocean Physics (JOHN R. APEL, PH.D.) (Halaman 186-195)