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					Icicle spacing
Thomas E. Lockhart and Tim Kline

Two theories 1,2 have been proposed to account for the spacing between adjacent icicles.
Both assume that the icicle spacing is determined by the spacing of liquid water drops
that then freeze.

I. Figure out the spacing by balancing the drop’s weight with surface tension.1 This
assumes that the drops individually achieve a certain size (and therefore, shape)
independently of each other. The spacing is then the center-to-center spacing of adjacent
drops, which is equivalent to the diameter of each drop measured at the substrate.
Makkonen and Fujii 1 calculate the spacing for an assumed drop shape to be 2.13 cm,
which generally matched their experimental values.

II. Attribute the spacing to a Rayleigh-Taylor instability2, which requires solving the fluid
dynamics equations (Navier-Stokes equations) for a film of liquid water. The Rayleigh-
Taylor instability occurs anytime a heavier fluid is located above a lighter fluid. The
heavier fluid falls down into the lighter fluid, with a characteristic rippling of the
interface between the fluids. One particular wavelength of ripple grows faster than the
others, and gives rise to the drop spacing. This approach incorporates the interaction
between the adjacent drops. de Bruyn 2 reports the spacing wavelength to be 2.47 cm, in
slight disagreement with method I. We won’t be pursuing the mathematical development
of this model due to its greater complexity (5 coupled partial differential equations).
However, we should note that this analysis incorporates the interactions between the
drops which are ignored in method I.

Nothing prevents the drop spacing from exceeding these values, because water may
accumulate on a surface at points determined by the surface itself or any channeling of
water leading to the icicle-producing surface, for example.

Our data indicates that icicle spacings are typically in the range from 1.6 cm to 2.3 cm,
with even larger spacings occurring in decreasing frequency.
We can account for this spacing range using method I alone, although we adopt a slightly
different procedure.3 Rather than using force balance, mechanical equilibrium for the
drop is expressed in terms of pressure, which leads to the Laplace form for pressure in a
curved surface. The Laplace pressure occurs in any situation where a tensile force is
balancing out a pressure, such as in our drops or in a balloon being blown up. This makes
a better estimate of the spacing by allowing the drop shape and size change as each drop
grows larger, so that the icicle spacing is determined by the drop profile at the time of
freezing. For the largest drops, the surface curvature has to be large (small radius of
curvature) in order to generate a bigger Laplace pressure. Counter to one’s intuition, the
drops get closer together as their individual volumes increase.
Icicle ripples
A different instability gives rise to a set of ripples running along the length of any one
icicle.4,5 These ripples have a very regular spacing (about 1 cm) which doesn’t change as
the icicle ages, but the amplitude of the ripples can change. This rippling is again due to
competition between two processes. Any little initial disturbance of a liquid film running
down the outside of the icicle will cause a bulge and an accompanying concavity. The
bulge’s convex surface will cool more quickly and is able to more rapidly dissipate the
latent heat of fusion upon freezing than the concave region (this is called a Laplace
instability). So water is more likely to freeze on the convex sections than on the concave
ones, which means the bulges build up more rapidly than the spaces between them. The
effect that counterbalances this is the flow of the liquid film itself which tries to even out
the temperature differences caused by the convexities and concavities. The two effects
come to equilibrium at a particular spacing between bulges: 0.8 cm or so, depending on
thickness of the liquid water layer.4

Details of the icicle spacing problem:
Method I: Drop shape determines spacing

Water drops hanging from a surface are called “pendant drops”. Their shape is
determined by a competition between the weight of the drop causing it to fall and the
surface tension that resists changes in the surface area of the drop.

According to Makkonen and Fujii,1 a pendant drop is supported by the force at it rim due
to the vertical component of surface tension:

(1) F,v = 2  r  sin 
                                                                                                          r
where r is the radius of the
                                                                               dr
drop’s base (where it contacts
the substrate),  is the surface                                          dz ds
tension and  is shown in the
accompanying figure 1.
                                                                             Figure 1: the pendant drop
                                                                             and its coordinate system
Following the same model,
each section of thickness dr                                       z
on the rim of the drop feels a
force of gravity given by

                  r  dr
(2) Fg  - ρg             2π r z dr
              r



where  is the density and g is the acceleration due to gravity.
Makkonen and Fujii then set F,v = Fg, and take a derivative of both sides with respect to
r, which gives

(3) 2 d(r sin )/dr = -2 g rz or

In figure 1, note that sin  = dz/ds and tan  = dz/dr. If we make the same small angle
approximation as Makkonen and Fujii so that sin   tan  (i.e., the very initial stages
when the drops are still almost flat), we end up with

(4) d(r dz/dr)/dr + grz = 0.

This can be rewritten in terms of the following dimensionless variables:

           γ
(5) λ       ;   r/;   z/
          ρg
 is often called the capillary length and sets the scale over which capillary forces are
important. For water at 0°C,  = 75.6 dyn/cm,  = 1.00 g/cm3, and g = 980 cm/s2, so  =
= 0.2777 cm.

Equation 4 then becomes
    d 2 1 d
(6)               0
    d 2  d
which is the zero-order Bessel equation of the first kind. Boundary conditions for this
problem are:

     dη
(7)      0 at ξ  0 and at ξ  α
     dξ
where  is the dimensionless radius of the drop’s rim. So the small angle approximation
leads to a drop in the shape of a Bessel function. To find , you have to find the first
nonzero value for which Bessel function  = J0() has an extremum (i.e, a minimum in
this case). This happens at  =  = 3.83 (Thanks, Lyle!!!).

The drop spacing will be twice the drop radius, so S = 2 = 2.13 cm.

Not surprisingly, this isn’t what we found in our data on icicles in nature. The reason is
simple: the small angle approximation effectively eliminates any effect of curvature
because the drop is considered nearly flat. Our data suggests that this is not the case with
icicles formed in nature. The discrepancy can be resolved by altering the procedure to
include more fully the role played by surface tension and surface curvature.
To do this, we have to follow the procedure outlined by Boucher and Evans.3 Rather than
explicitly consider forces, we will look at pressures. The mechanical equilibrium
condition for the interface between two fluids is given by the Laplace formula:

(8) p = (1/R1 + 1/R2)

where p is the pressure difference between the two fluid, is the surface tension and the
R’s are the principal radii of curvature for the interface. Using the coordinate system
shown in Figure 2 (note that z is reversed from figure 1), we note that R will be a
function of z. If we treat the air pressure as negligible compared to the liquid pressure, p
= p, where p is the liquid’s pressure. In any fluid, the pressure at a depth is caused by the
weight of the fluid above it, so p = g(h - z), where we have set the density of the air
below the drop equal to zero and h is the maximum displacement of the drop below the
substrate.
                                                                  z
The Laplace equation                                          h
becomes:
                                                                                                dr
(9) g(h-z)(1/R1 + 1/R2)
                                                                                         dz     
R1 and R2 are functions of x                                                                     ds
and z, so this equation                                                      
determines the actual drop                                                                            r
                                     Figure 2: the pendant drop
shape. At any point (r, z) on
                                     coordinate system for the
the drop’s surface, the tangent      Laplace calculation
to the surface makes angle 
with horizontal. We define s
as the total distance along the profile from (0,0) to (r,z). Therefore dz/dr = tan  and cos
 = dr/ds. The radii of curvature at that point are related to :3

(10) R1 = dr/d(sin) = dr/(cos  d), R2 = r/sin

Put all this together in equation 9 and we get

(11) g(h-z)(d/ds + (sin )/r)

Solving this for d/ds gives:

(12) d/ds = g(h-z)/ - (sin )/r

In addition, there are some boundary conditions: s = 0 and  = 0 at (0,0). In addition,
dr/ds = cos and dz/ds = sin . Now in principle, all we have to do is integrate this for
each point on the profile (each s value) and we’d know the profile. The trouble is there is
no closed form solution and the integral must be done numerically. Rather repeat the
calculations, we’ll defer to Boucher and Evans3, who determine a number of profiles,
corresponding to different drop volumes. Their results are scaled by a factor a such that
a2 = 2/g, which means a  2 λ in the analysis of Makkonen and Fujii1. In particular,
Boucher and Evans generated a table for the radius, depth and volume of drops below an
infinite plane. One of the primary results of their numerical calculation is the existence of
a maximum possible drop volume. The radius of such a drop is 2.2766 a, which
corresponds to a drop spacing (assumed to be equal to the drop diameter) of 1.79 cm at 0
°C. The other interesting feature is that for smaller volume drops, the curvature is such
that the drop radius at its base is larger than that of the maximum volume drop, with a
maximum radius of 2.71 a, corresponding to a diameter of 2.13 cm. In other words, small
volume drops with very small h values give a result exactly the same as Makkonen and
Fujii. This isn’t surprising, because the small angle approximation they used is the same
are requiring a small  (and hence a tiny h value) in the Boucher and Evan calculation.
Presumably, one could get exactly the same result by using equation 3 rather than the
approximations that led to equation 4.

What is perhaps a bit disturbing is that nothing in this calculation takes the interaction
between adjacent drops into account, and yet the spacing can be maintained over quite
large distances. Something has to enforce this spacing between drops and yet the model
calculates the size of each drop individually, independent of all the others in the array of
drops. A moment’s reflection may give a clue: this is analogous to the spacing of birds
perching on a wire.

Each bird stays just out of reach of
the neighbors, so a minimum
spacing is enforced just by requiring
the birds to all coexist on the same
wire. Similarly, the water drops get
as big as they can before freezing,
running out of water, or falling off,
resulting in a more or less uniform
spacing but not less than the
minimum spacing required by the
drop size.
References:
1. Makkonen, Lasse and Fujii, Yutaka “Spacing of icicles”, Cold Regions Science and
   Technology 21, 317-322 (1993).
2. de Bruyn, John R., “On the formation of periodic arrays of icicles”, Cold Regions
   Science and Technology 25, 225-229 (1997).
3. Boucher, E.A. and Evans, M.J.B., “Pendent drop profiles and related capillary
   phenomena”, Proceedings of the Royal Society London A 346, 349-374 (1975).
4. Ogawa, Naohisa and Furukawa, Yoshinori, “Surface instability of icicles”, Physical
   Review E 66, 041202 (2002)
5. Ueno, K., “Pattern formation in crystal growth under parabolic shear flow”, Physical
   Review E 68, 021063 (2003)

				
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