GFD-1   Problem Set 2    OC-512/AS-509

out: 17 Jan 2001

back: 24 Jan 2001  

 

1.   (a) An ice-cube floats in a glass of fresh water. As it melts does the free surface rise, fall or stay the same?  [think beyond the first, most obvious answer].

      (b) If the glass of water is on a rotating table, with the water ‘spun-up’ (that is rotating with the table), does the ice cube begin to rotate as it melts?  If so, which way? First suppose the water is fresh. Then suppose it is salty.

 

2.  An isothermal, ideal-gas, dry atmosphere,  with temperature 250C, initially motionless, is heated uniformly by an amount 50C.

 (a) Calculate the change in available potential energy, PE, of an air column

                

                                              

the latter using our approximation to the geopotential F @ gz. You will want to use the constraint that the mass of the air column does not change.

(b) The sum PE + E, where E is the internal thermodynamics energy (here CvT) is equal to the heat added (1st law of thermodynamics).  What is the partition dPE/dE between internal energy change dE and (external, mechanical) potential energy change dPE?

 

(c) How far upward does the center of gravity of the air column move due to the heating?

 

(d)  The top 10m of the tropical ocean, initially isothermal, (T=T0 = 250C, salinity S = 35 parts per thousand, pressure close to atmospheric)  is heated by 50C.  Repeat the above calculations. Use a linear, approximate equation of state  r = r0(1 - a(T-T0)) where a and r0  are found in Gill, Appendix 3 p603­.  Thus work with the integral

                                                                       

                                                        

where for convenience we set z=0 to be a fixed level and z=h to be the top of the ocean. Initially, before the extra heating, h = 10m.   Note once again that to conserve mass, does not change; here with uniform density in that layer the integral is just hr.

(e)  If, in these examples, some process allowed the change in gravitational potential energy due to this heating event to be converted to kinetic energy,  , what would the typical change in horizontal speed, U, of air and water be?

 

Do these answers tell us anything about the overall energetics of ocean and atmosphere?

3. Take one of the laboratory demonstrations (written up on the class web-site, www.ocean.washington.edu/courses/oc512/gfd1.html) and write a page (single-spaced) of discussion, adding some quantitative notes.  Thus,  for example you saw that when two fluids of temperature roughly 300C and 50C were mixed, the volume of the mixture was reduced, owing to the curvature of the equation of state.  Using the plotted equation of state, what should the fractional volume change have been.  Or, as a second example, we saw gravity waves being stopped by an adverse current.  This must occur where the wave’s group velocity is equal and opposite to the current speed; using the deep-water dispersion relation, and observed wavelength, what should that adverse current speed have been?

4. Show, using scale analysis that the hydrostatic approximation for long gravity waves (with uniform density r) is valid if (H/L)2 << 1.  (Note that this is ‘easier to achieve’ than H/L << 1).

MOM                            

MASS                           

 

 To do this it is not enough to show that the vertical acceleration is much less than g in the vertical MOM equation; what you must show is that the error is small in making the substitution shown in the x-MOM equation, first line.  [One can take the general solution for short/long gravity waves and show when the hydrostatic limit is valid, but here we suggest using scale analysis.]  The essence of long waves is in the horizontal acceleration and horizontal pressure gradients which are overseen by the hydrostatic vertical balance. So, we want to show that

 g h/x is an accurate expression for 1/r p/x.

The former is independent of z, yet the latter is not.  The variation in z of p/x from top to bottom  is (to the level of scale analysis)  ~ H2p/xz which we want to be small compared with 1. Use the 2d MOM equation and the MASS conservation equation to derive the desired result.

  Notes about scale analysis:  there are two related techniques in dynamics, dimensional analysis and scale analysis.  With dimensional analysis we formally define new variables which have no dimensions, like x’ = x/L,  u’ = u/U, using constants L and U that carry the dimensions and usually are related to imposed scales like the radius of a cylindrical obstacle (L) or the upstream mean flow speed (U).   After substituting these in the equations of motion, this leads to non-dimensional equations  which show you the key parameters describing the flow, and give you modelling principles.  For example, a model airplane in a wind-tunnel will cause exactly the same streamline pattern as a full-size airplane if the Reynolds numbers of the two flows are the same (but, pressure, length and time are all distorted in amplitude, comparing one flow with the other).

   Scale analysis, on the other hand, is simply the estimation of typical sizes of terms , using observed length, time, velocity scales L,T,U; U is the typical size of the x-velocity component, u.  The length scale L of a velocity field is the typical size of, say,  U/(u/x).  For a sinusoidal field,  u = A sin kx, hence  U/(u/x) = (1/k) tan kx and we ignore the tan(), concluding L ~ 1/k  (we are not interested in places where u/x is very small). Very often there are different length scales in horizontal and vertical, so we use two symbols L and H, respectively.  MASS conservation has the scale analysis  W/U~H/L in two dimensions. 

   Scale analysis can fail or mislead; for example the 3-dimensional MASS conservation,

                                             

would suggest W/U~H/L also,  if we suppose that u and v both have typical sizes U, and likewise for length scales in x and y. But,  we will find later that there can be close cancellation of the first two terms (‘horizontal non-divergence’) and often  W/U << H/L in atmosphere/ocean flows.

   Actually, there are three (related techniques). The third is called ‘coordinate  stretching’,  often ‘boundary layer stretching’.   In this case we have variations in dynamics in different parts of a flow.  With small viscosity, n,  a term like n 2u/z2   has typical size ~ nU/H2  yet the vertical length scale H is not likely to be that of the large-scale flow or wave.  The no-slip boundary condition at a rigid boundary causes u(z) to go to zero over a very short distance. This suggests that in that region a new ‘stretched coordinate’  z’ = z/d be defined, and d chosen so that the viscous term becomes comparable with the dominant terms in the horizontal momentum equation.  This leads to the mathematical technique known variously as boundary-layer theory,  multiple-scale analysis and ‘matched asymptotic expansion’.