GFD-1   Problem Set 4    OC-512/AS-509

out: 12-14 Feb 2001

back: 21 Feb 2001  

 

 

Problem 1. Viscous diffusion (see review of viscous stress on the class web page).  Solve the momentum equation

                                           (1)

 for the velocity field as a function of y and t, given that at the initial time, t=0,

                                    u = cos l0y;

there is no planetary rotation (f=0) and assume that there is no pressure gradient at all (this is a statement about there being no distant boundary at which pressure forces are being applied). Discussion. You can use the separation of variables technique, trying a solution form

                                    u = m(t) n(y)

where m and n are functions determined by substitution in the equation, or you can try guessing the form of the separable solution.  (1) is the classic diffusion equation or ‘heat equation’, since also describes the temperature field diffusing without any fluid motion.   Compare with a scale analysis prediction for the relation between time and length-scales.  This may seem a very idealized problem, but note that the equation is linear and hence you can add up many solutions for different values of l0, and get new solutions. Thus one can solve for the viscous destruction of a jet (say u = exp(-y2/a2)) by Fourier transform, which does this addition of many sin-waves.

 

Problem 2.  For linear Poincare’ waves (long, shallow-water gravity waves in a single layer of rotating fluid), suppose a given wave,

                        h = Real A exp(ik1x +il1y -iw1t)

encounters a rigid vertical wall lying at y = 0  (where the boundary condition is v = 0).  Find a reflected wave,

                        h= Real B exp(ik2x +il2y -iw2t)

that allows this boundary condition to be satisfied.  Discussion: Thus you want to write the v-velocity in terms of h (see Review Problem # 10  on website).    Note that we are given the wavenumbers and frequency of the incident wave, and must find those for the reflected wave, as well as the amplitude B.  k and w  can be found immediately by realizing that the two waves must cancel out along the boundary at y=0, at all times and all values of x; this suggests strongly that k2 = k1 and w2 = w1.  Another way to look at this is that all the interesting action is in the y direction and the x and t parts of the solution can be ‘separated out’, writing h = [some function of y] x  [exp(ikx -iwt)].  ‘Some function’ here expresses both incident and reflected waves.  So use the equation to solve for l2 and then use the boundary condition to solve for B as a function of A, k, w. Sketch h(y) at a couple of instants in time.

 

   This problem relates to the appearance of the Kelvin wave in our rotating long wave equations. Here is the idea. With f=0 (non-rotating long waves), the boundary condition on a vertical wall is simple: u= 0 for a wall lying along the y-axis, hence from x-MOM, h/x = 0 at x=0.  With this homogeneous boundary condition the reflected wave is a mirror image of the incident wave. Indeed, if an oscillating point source of waves exists to the right of the wall, at x = a, y = 0, the reflected waves can be found by simply placing an identical ‘image source’ of waves at x = -a, y = 0.  This ‘method of images’ fails when Coriolis effects are added in because there is a phase shift: look at your expression for B/A.   This turns out to require a new wave mode (Kelvin wave) when the oscillating wave-maker is near the boundary.  The same idea predicts edge waves of many kinds, in gfd situations where there are two active restoring forces (here, gravity and rotation), which lead to a mixed boundary condition rather than homogeneous Dirichlet or Neumann boundary conditions (like h = 0 or h/y=0 at y = 0).  

 

Problem 3.

    Solve for the velocity field during a viscous spin-down event, with initially geostrophic flow of an homogeneous fluid, r = const.  Take as a specific example the flow

                                     u = U0 cos (my) at t=0,

There is a rigid surface at z=0, with no viscous stress (a ‘free-slip’ boundary) and a rigid lower boundary at z = -H, where the Ekman layer develops.

            During the viscous spin-down make estimates of the distance travelled by fluid particles in the boundary layer and the interior. Draw some pictures of the ‘meridional circulation’ of these particles (that is, up-down/north-south streamlines and particle paths for the above initial flow).   Does every fluid particle pass through the boundary layer during spin-down?  For this problem assume that steady Ekman boundary layer theory applies (which means ignoring damped inertial oscillations near the beginning of the experiment).

 

 

Problem 4. 

            Write one single-spaced page describing a lab experiment from lab #5 (Ekman layers...).