Feb 15: Tides and tidal currents (3)

 

Depth-averaged momentum and continuity equations:

 

 

 

A widely used bottom drag parameterization is

 

               

 

where        is the magnitude of the near bottom velocity.

 

This is in analogy to the quadratic drag parameterization, that stems from a dimensional analysis, for a body with drag coefficient CD in a fluid stream of velocity UB and density r.

 

A typical value for the drag coefficient is  CD = 2 x 10-3 (dimensionless)

 

For a bottom current of 0.5 m s-1 this gives a stress of 0.5 N m-2 (or Pa).

 

 

Linearization of the momentum equations

 

The occurrence of non-linear friction is going to make simple analysis difficult.

 

An alterative parameterization is a linear drag law: 

 

     where r has dimensions of velocity.

 

Bowden explains (rather glibly) the rationale behind introducing the depth average velocity and the relationship between r and CD.

 

The parameterization relies on the velocity profile maintaining a constant shape.  This is generally a reasonable assumption.

 

Observations of the tidally reversing flow in the Hudson River show this behavior.

 

To use this parameterization, r may need to be tuned to reflect the typical magnitude of the near bottom velocity w.r.t. the depth-average.

 

By removing the dependence on the actual time-varying velocity we have removed the nonlinearity of the friction from the equations.

 

Before we can proceed with a linear analysis of the 2-D depth averaged equations, we need to consider the non-linear momentum advection terms.

 

These are of order U2/L, while the Coriolis term is of order Uf  (adopting the inertial period as the characteristic time scale).

 

The ratio of non-linear to Coriolis terms is U/fL = the Rossby number.

 

For tidal currents on the shelf of order 0.5 m/s and an along-shelf length scale of L = 100 km

 

           U/fL = 0.5/(10-4 x 10 x 103) = 5 x 10-2

 

so we can be reasonably justified in ignoring the nonlinear terms (for now).

 

 

A third non-linearity arises from the occurrence of terms involving .

 

We will limit the situations we consider to cases where the tidal height variation is modest compared with the water depth, i.e.

 

Then we get the linearized 2-D depth-averaged equations for tidal flow:

 

           

 

           

 

Since these equations are linear, they could be solved for each harmonic constituent independently, and the various solutions superimposed to give the response.

 

Progressive waves

 

To examine the dynamical processes that these equations describe, it is instructive to consider the simple unforced case:

 

How do the sea surface and currents respond if the ocean is distorted in some way, but the disturbing force is removed?

 

The response is in the form of free waves.

 

Omit friction at first:

 

           

 

A solution can be obtained for the case of a wave traveling in the x direction with no movement in the y direction, i.e. v=0.

 

                   

 

Differentiate (1) w.r.t. x, and (3) w.r.t. t, and difference them to get

 

                

 

This is a wave equation with wave speed given by c2= gh.

 

The general solution is:

 

           

 

from which it follows that

 

               from equation (1).

 

Equation (2) shows that

 

           

 

The amplitude of the wave decreases exponentially in the y direction.

 

The length scale of the exponential decay is

 

which we see can be interpreted as the distance the Kelvin wave (or any long wave in shallow water) travels in one inertial period.

 

[Sketch schematic of  Kelvin wave - Bowden fig 2.7]

 

Amplitude decreases to the left of the direction the Kelvin wave propagates.

 

Exponential term means the Kelvin wave must be limited by a coast (where the pressure gradient can be balanced by the coastal boundary)

 

Simple harmonic wave form: 

where k = wavenumber , and

 

In the deep ocean

 

·        h = 4000 m, g = 9.81 gives c = 200 m/s

·        c = wavelength/period for M2 = 12.4 hours

·         => wavelength = 8850 km

·        R = 2720 km (at latitude 30o where f = 7.29 x 10-5 s-1)

·        For e.g.  A = ~0.5 m   we would get  U = ~2.5 cm/s

 

 

Frictional effects on waves

 

 

Eliminating  from (1) and (3) gives

 

 

which is a wave equation with a damping term.

 

Anticipating that the same form of solution will work provided we accommodate the possibility the wave amplitude decays as the wave propagates, we try:

 

 

Differentiating and substituting into the damped wave equation we find (by independently satisfying the real and imaginary parts of the equation) that:

 

or  

 

showing the wave speed is reduced.

 

We also so have that:

 

 

so after dividing through by 2k we estimate that frictional decay scale

 

 

provided we assume c/co is approximately equal to 1 (i.e. friction is weak).

 

The solution for velocity can be derived from continuity

 

 

after assuming a solution of the form

 

 

which incorporates the likelihood of a phase shift with respect to the sea level displacement:

 

 

The solution is:

 

 

 

Notice that the current:

 

·        is no longer in phase with the amplitude

o       the current peaks a time a/w before the sea level crest

·        the magnitude is reduced compared to the same frictionless wave