Wave modification by currents .1 Introduction .1 Introduction

In the nearshore region, the most important currents are horizontal and they have horizontal extensions (length scales) that are of the order of a wave length or much larger. Due to turbulence from various sources and the mechanisms responsible for generating the currents, they may show vertical variations in the horizontal velocity, typically in magnitude, but often the direction of the velocity is also varying over the vertical.

3.6 Wave modification by currents 155

The presence of currents changes the waves. However, as with depth refraction, the lowest order effect turns out to be in the dispersion relation through changes in the speed of wave propagation. Because those kinematic properties do not depend on the wave amplitude] we are, in parallel with the depth refraction] able to analyze the propagation patterns in linear waves on currents separately from the analysis of the dynamics which provides the information about the amplitude variations.

A common assumption is also that we analyze the effect the current has on the waves but assume the currents remain unchanged the waves.

This of course is a simplification that is discussed extensively in the chapters about wave induced currents and nearshore circulation.

In the following, we first analyze the local effects of the simplest case, a wave motion on a steady current, uniform both over depth and in the hori- zontal plane. Then give a brief overview of the local effects of depth varying currents] and finally briefly discuss extension of the geometric optics ap- proximation t o the case of combined wave-current motion in the horizontal plane.

3.6.2 Waves o n a steady, locally u n i f o r m current

Thus, consider a current with the steady velocity U. The uniformity of the current means an infinitely large length scale. The primary effect is that the wave will propagate relative to the moving water.

Therefore] ^a wave with wave number k will satisfy the same dispersion relation found earlier, that is

w, = tanh k h (3.6.1)

where the relative wave frequency] represents the frequency mea- sured by an observer moving with the current velocity and k In the general case, the current direction will differ from the wave direction. The wave speed, however, is measured in the direction perpendicular t o the wave front. Therefore] the current component parallel to the wave front will not contribute to the speed of the wave. From a coordinate system fixed in space, we will therefore see the wave moving at an absolute speed

which is in the direction of k and given

+

^k

.

U (3.6.2)

156 Introduction t o nearshore hydrodynamics

or

=

+

(3.6.3)

where U and is the angle between k and U. The situation is illustrated in Fig. 3.6.1.

However, since the wave is merely translated with the current, the wave length will look the same from the fixed and moving coordinate system,

is the same. Hence, we will have, since w =

W, = W, = (3.6.4)

which substituted in (3.6.3) gives the relation

= w,

+

^(3.6.5)

This is also called the Doppler relation. It relates the absolute frequency w, observed from a fixed coordinate system to the relative frequency w, which is the frequency that satisfies the dispersion relation (3.6.1) (and which is observed from a coordinate system moving with the currents.

Fig. 3.6.1

steady uniform current (from Jonsson, 1989).

The relation between absolute and relative phase velocity for a wave in a

Notice that for a monochromatic wave motion wave fronts will be con- served, so that in any region the number of waves moving into the region will be the same as the number of waves moving out of the region. This

3.6 Wave modification b y currents 157

means that the absolute wave period and hence the absolute frequency w, are conserved. That is also the frequency that in practice is easy to measure through measurements of the absolute wave period =

the period observed from a fixed point in space. On the other hand, to evaluate other features of the wave motion such as velocities, pressures, etc., we need t o determine through w,. Therefore, the solution of (3.6.5) is important.

Substituting (3.6.1) into (3.6.5) we get

w, ^- cos = tanh k h (3.6.6)

which shows that the solution kh for given w, depends on cos which therefore is the parameter for the problem.

While a numerical approach is required for specific cases, we see that solving (3.6.6) essentially corresponds t o seeking crossing points between the two curves.

= w, -

(3.6.7)

=

This is illustrated in Fig. 3.6.2.

In the figure, = 0 corresponds to w, = (point E in the figure), a following discussion we assume that the current is changing slowly enough that locally it can be considered uniform. If a wave motion moves from a region with no current (point E) into a region with increasingly stronger opposing current the line for w, - cos gets steeper and point A moves upwards on the upper curve in the figure. Clearly this means that the stronger the opposing currents the larger the wave number becomes, that is, for a given w, opposing currents reduce the wave length.

In this general case of a weaker current, A corresponds to a situation where both the phase speed c and the group velocity (that is energy prop- agation velocity) ⁼ are

>[

cosp Thus, both the wave train

158 Introduction t o nearshore hydrodynamics

k 0

Fig. 3. 2 Solutions the wave number in terms of for given values of h and Ucosp. Notice that can be both positive and negative. The case of following currents corresponds to > 0 (upper branch of - while opposing currents correspond t o cos < 0 (from Jonsson, 1989).

with this and its associated energy are able to move against the current, a group of waves will propagate upstream.

When the current gets strong enough cos numerically large, the straight line in the figure gets steeper) the waves are blocked by the currents from propagating further. In the figure this means A reaches point F (one solution) which is the blocking point or caustic point. Point F represents waves just able to maintain their position against the current because at F the absolute group velocity = 0 (Jonsson et al., 1970). Since the group velocity is defined as it is obtained by differentiating (3.6.5) with respect to This gives

=

+

^(3.6.8)

Thus blocking of the waves occur when =

3.6 Wave b y currents 159

Figure 3.6.2 is essentially based on ray theory, and if an analysis of the wave height variation approaching the blocking point is carried out according to that theory it results in infinite values (the wave height is singular). However, Peregrine (1976) conducted a nonlinear analysis in the neighborhood of the blocking point and showed that in reality the waves have large but finite steepness. It also turns out that (if no breaking occurs) the waves are reflected at the blocking point but with a change in wave number. The wave solution at point C, the second solution, corresponds to the waves reflected from the blocking point. Since the early contributions quoted above this problem has been discussed by a number of authors and lately by and Schaffer and by Chawla and Kirby (2000).

In nature blocking frequently occurs outside inlets and in river mouths with strong (often tidally dominated) currents. Where the current is strong enough to arrest the waves, the result is a region with very waves that often break before they reach the actual point. Such regions represent navigational hazards particularly to smaller craft.

If the opposing current gets even stronger (the entirely above the there are no solutions, no waves are able to propagate up against the current.

Following currents

The case of Ucosp

>

0 corresponds to currents following the wave motion. It is represented by the lower branch of - cos in Fig. 3.6.2.

Here, solution B represents the simple case of a wave motion on a following current. It is seen that B corresponds t o a kh-value smaller than for case E (wave with the same period on no current) indicating the effect of the following current is to increase and hence L.

Wave orthogonals and wave rays

In waves with currents the wave orthogonals and the wave rays generally have different directions.

The wave orthogonals are defined as curves normal to the wave fronts.

As Fig. 3.6.1 shows in a (locally) uniform current the absolute phase ve- locity given by (3.6.1) is in the direction of the wave orthogonal.

A wave ray, on the other hand, is defined as the curve along which wave energy is moving. For a fixed system this will be in the direction of the absolute group velocity The definition of (and thereby the direction of the ray) is given by

C g a = Cgr

+ u

^(3.6.9)

160 Introduction to nearshore hydrodynamics

which differs from the definition (3.6.3) for and has a direction different from the orthogonal. When the wave height varies along the wave front then the current component along the front matters because in addition to moving energy in the direction of the orthogonal it creates a net energy flux along the front.

3.6.3 Vertically varying currents

The solution of the Doppler version (3.6.5) of the dispersion relation represents a solution of the local problem of determining the parameters required to evaluate the formulas for velocity, pressure, etc. for the wave motion a t a point.

Similarly, in this section we discuss the problem of determining the modification of the wave motion at a point in various cases with currents that vary over depth. This problem has been discussed in the literature by various authors. For an extensive overview of the dispersion relations for various cases see Peregrine (1976). An account of the solution for some simple current profiles is given by Dingemans while an approximate solution for more general current profiles is found in Kirby and Chen (1989).

The following is a brief overview, which combines the last two versions. It also also provides some information about the changes in the wave particle velocities. For simplicity the discussion is limited to 2DV motion.

The analytical solutions available all assume that the current profile is known from measurements). The horizontal variations of the current field are assumed so slow that contribution to the equations from horizontal variations of the current can be neglected in the local relations.

On the other hand, the current velocity is assumed relatively strong in comparison to the wave particle velocity w, the horizontal, the vertical component, respectively, of the wave motion.)

Thus, we consider a surface elevation and total velocities w that be written

=

+

^{t ) (3.6.11)}

=

+

^{t ) (3.6.12)}

3.6 Wave by currents 161

Here it is assumed that the wave amplitude is small enough that

w, = that = = 0, and that = =

where are small parameters measuring the magnitude of the wave ve- locities and the gradient of the current, respectively.

Substituted into the inviscid fully nonlinear (that is Euler) equations of motion and the associated (nonlinear) boundary conditions, this results in the following lowest order of the equations for inviscid motion.

(3.6.13)

(3.6.14)

(3.6.15) where is the dynamic pressure created by the wave motion. We see that these equations are linearized because we assume u,. The boundary conditions are

=

-

_at

+

^{z = o (3.6.16)}

= 0 z

z = o (3.6.18)

which are linearized in a similar way as the Euler

We assume the wave part of the motion is described by a stream function

- (3.6.19)

which implies that

a$ . ,

^-- _(3.6.20)

and where and are also slow functions of xi and t. Substitution of this into the equations for the wave motion then shows that must satisfy

we seek only information about the effect the current has on the waves, but assume the currents unchanged, these equations assume inviscid motion also do not contain the (viscid or turbulent) terms required to actually maintain the assumed current profiles over some horizontal distance.

162 Introduction to nearshore hydrodynamics

the equation.

(3.6.21)

In document Introduction to Nearshore Hydrodynamics (Page 177-185)