here is a picture of a Tesla coil with a discharge that seemingly terminates in the air
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What completes the circuit at the end of the streamers?
[Physics] Tesla coil arc in air – where is it going
electricityelectrostatics
Related Solutions
Here is an approximation of what you are trying to do. The magnetic field inside a solenoid is given by (from Wikipedia) $B={\mu}nI$ where ${\mu}$ is the permeability of the medium (presumably water in your case), $n$ is the number of turns per unit axial length, and $I$ is the current. The maximum value of $n$ is determined by the thickness of the wire you are using. This vendor sells magnet wire as thick as 12 gauge, which for a single layer wrapping yields $n=468$. The permeability of water is very close to that of vacuum, so use ${\mu}=4{\pi}{\cdot}10^{-7}$ $H{\cdot}m^{-1}$.
Achieving your minimum desired field of 0.1 Tesla requires a 170 Amp current which is probably well beyond the safe operating range for 12 gauge wire. Now this is an engineering problem. You could look for a thicker wire and also wrap multiple layers to reduce the required current. If the water is flowing that will help keep the wire cool, but being submerged demands careful consideration of electrical isolation. Here are some epoxies for potting electronics, though I suspect there are many other options available. Hopefully the links provided help illuminate what keywords to use in your research.
Once you have an idea of what kind of wire to use (and its resistance), and also the max current you'd like to run, you can use $P=I^2R$ to estimate the power you will need. Then you can start shopping for a DC power supply.
A 1 Tesla field is huge!
To the senior members: I apologize in advance if it was in bad taste to link to vendors.
The conventional definition of lightning is a current though a plasma (not necessarily through air as lightning happens on other planets) so in a vacuum there cannot be any lightning. However charge does still flow between electrodes in a vacuum, and from personal experience I know that we can get something very like lightning in the right circumstances.
In a vacuum charge can leave a surface due to field emission. Basically the field gets so strong that the electrons at the surface can get enough energy to overcome the metal work function by moving only a small distance and they tunnel out of the metal. However this is a rather diffuse flow and wouldn't generate any light so it wouldn't resemble lightning.
I did claim you can get an effect like lightning, and this can happen if there is a rough spot on the electrode surface that concentrates the electron flow. If the electron flow gets concentrated enough it can vaporise the electrode at the high point then you get an arc through the vapour. In the old days when we used giant Van de Graaff generators as particle accelerators this used to happen on a regular basis leaving a scar on the dome that then needed to be polished away.
Best Answer
What the picture shows is a corona discharge (see also Wikipedia).
It isn't a circuit in the usual sense of the word. It happens because the voltage is so high that it raises the electron energy to above the work function and the electrons just leak off. In effect the coil is charging the air around it. The charge will end up on the furniture, walls, floor, and indeed you if you're in the vicinity.