Since I have swam on the swim team for most of my life, I am very familiar with bubbles. I know a raindrop falling through the sky gets its shape because it is the most aerodynamic shape, but how come a bubble rising through water has a different, rather jellyfish-like or dome shape? is an example of what I am referencing. (As a side note, I have found this property to be shown mostly in larger bubbles.)
As the bubble rises it pushes the water above it out of the way, so we get a water flow created around the bubble. With a large bubble the flow velocities will be relatively high because a large bubble has to push aside a large volume of water.
Water flowing around a bubble will pull it out of shape. the obvious simple example is a bubble in a shear flow, that gets pulled out into an ellipsoid:
For a bubble rising through water the flow is going to be complicated and there is no simple way to calculate it. We have to reach for a finite element analysis program and a large computer. I did some Googling and managed to find this paper reporting calculations of this type$^1$. If you look at for example figure 3 in the paper it shows the hemisphere bubble shape that you show in your photo.
The trouble with the computer modelling is that it can be hard to get an intuitive feel for what is going on, so I've attempted to draw my own diagram of how the bubble is pulled out of shape by the water flow:
You shouldn't take this too literally as it's just an illustration of the flow. The arrows show the water flow, and how it pulls the bubble out of shape. There are more detailed (if more confusing!) diagras of the flow in figure 15 of the paper I have linked.
$^1$ Jinsong Hua and Jing Lou, Numerical simulation of bubble rising in viscous liquid, Journal of Computational Physics 222 (2007) 769–795
The bubble maintains a spherical shape, because of surface tension, which I am sure you are aware of. It does so , because it wants to store maximum volume of the fluid involved, using minimum surface area, which can be done using a spherical shape. Thus, the layer of the bubble acts like a stretched membrane. When the pressure inside the bubble, for whatever reason, increases more than what the surface tension can balance, it breaks apart into a semi-spherical, or dome like structure. Now, even though it is not a complete bubble, it still has surface tension, and thus possesses some concavity.
The reason behind the bubble's desire to minimise it's surface area, is that all objects want to stay at the lowest energy possible. When you increase the surface area of the bubble, you increase the number of particles being affected by adhesive forces of the surrounding, and thus increasing it's energy( surface potential energy).