Why will two bubbles floating on water surface attract each other? Two identical bubbles floating on water surface will form clumps, according to the "cheerio effect". But what's the detail about the force? It's necessary to calculate the shape of water surface, in order to find the force?
 A: Vella and Mahadevan  explain the effect as follows:

For simplicity, we consider
  the latter case schematically illustrated in Fig. 2, although
  the explanation of the clustering of many bubbles is similar.
  Here, the air–water interface is significantly distorted by the
  presence of the wall the well-known meniscus effect, and
  because the bubble is buoyant, there is a net upward force
  due to gravity, Fg , on the bubble. Because it is constrained
  to lie at the interface, however, the bubble cannot simply rise
  vertically, and instead does the next best thing by moving
  upward along the meniscus. [..........]
  A single bubble will deform the interface just as the presence
  of a wall does, although for a different reason and to a
  lesser extent. In the case of the bubble, it can only remain at
  the interface because the buoyancy force, which tends to
  push the bubble out of the liquid, is counterbalanced by the
  surface tension force, which opposes the deformation of the
  interface and hence acts to keep the bubble in the liquid.
  These two competing effects reach a compromise where the
  bubble is partially out of the liquid but the interface is
  slightly deformed. This deformation is sufficiently significant
  to influence other bubbles nearby, which move upward along
  the meniscus and so spontaneously aggregate.


They also provide an answer for the force of interaction between two particles with a Radius $R$:
$$F(l)=-2 \pi R B^{5/2}\Sigma^2 K_1\left(\frac{l}{L_c}\right)$$
With Bond number $B= \frac{R^2}{L^2_c}$, $L_c=\sqrt{\gamma/\rho g}$, $\Sigma$ a dimensionless Archimedes weight parameter and $K_1$ a first order Bessel function.
For more information, consult the paper.
A: Bubbles don't just attract on the surface of water but also under water below the surface. Bubbles attract because they emit / produce ultrasonic waves which leads to the Bjerknes attraction phenomenon. Read 'The Acoustic Bubble' by Timothy Leighton. Gerald Pollack's 'The Fourth Phase of Water' also provides some good information, but basically, bubbles attract because of mechanical ultrasonic vibrations.
