The spell Globe of Tranquil Water creates a bubble of water with a 20-ft radius around the caster and moves when the caster moves.
Is the caster able to swim through the orb, implying that the caster can effectively swim through the air or be launched into orbit via the spell Buoyancy?
Note: This answer challenges the question's frame, reading the spell in a nonstandard way.
Unless there's already water around the caster when the caster casts the spell globe of tranquil water, there's no water inside the effect created by the spell globe of tranquil water. Bear with me! Here's the spell's description:
Upon casting this spell, a rippling bubble of calm water extends outward from you to a radius of 20 feet and remains centered on you when you move. The bubble blocks all natural and magical precipitation such as rain, snow, and hail (including spells such as ice storm). This bubble also pushes out any fog or mist within the area, though it does not affect temperature and can’t block natural or magical lightning.
(Emphasis mine.) The problem with this description is that the emphasized sentence makes the reader—and that included me!—think that the bubble's area should be filled with water. (Seriously, I, too, imagined the spell's effect as if the caster were some kind of aquatic hamster in a water-filled hamster ball.)1 But that first line describes visually what happens when the spell's cast: That bubble that's made of water starts around the caster and moves outward from the caster then surrounds the caster at a distance of 20 ft. However, nothing in the spell's description says that the bubble's interior is filled with water!
Thus, when the spell's cast, what's closer to a shell made of tranquil water billows forth from the caster, pushing away fog or mist, and stops. The area between caster and bubble is otherwise unaffected. (The GM could, I guess, rule that the expanding bubble leaves the area damp.) The remainder of the spell assumes this bubble's in place: it stops precipitation (but temperature and lightning still get through), and the formation of the barrier pushes fog and mist out of the area and the barrier keeps out that fog or mist while the caster travels. (This fog-and-mist-pushing effect of what's otherwise a barrier—the spell's of the abjuration school—is kind of important as it makes the spell a great defense against spells like cloudkill and solid fog, which would otherwise see their effects trapped inside the barrier with the caster were the spell a normal area spell!)
Then the description continues with this:
All water in this radius counts as calm water for the purposes of Swim checks, and the bubble prevents sprays and blasts of mundane and magic water from striking with enough force to deal damage, move creatures or halt their movement, or perform any action that requires an attack roll or combat maneuver check.
Thus, while the globe effect remains, the caster and his buddies can be on land and struck by a tidal wave, yet the globe effect means all that water that would normally shove them around or deal them damage is, instead, rendered harmless. (The caster and his buddies will still get wet, though—a tidal wave is not precipitation!). Similarly, the caster can leap off a boat into the sea during a raging thunderstorm, and the globe effect means the water that now surrounds him is calm and easily traversed.
Reading the spell globe of tranquil water this way—as a barrier that turns any water that gets within into tranquil rather than as a sphere that's filled with tranquil water—makes the spell into a reasonable niche spell, usable as a defense against casters fond of battlefield control spells, instead of a spell that's capable of, for example, transporting an army through the air so long as the caster's arms don't tire or drowning a structure's occupants by having some unseen servants open the doors.
Hence a caster that wants a spell that allows an aquatic creature to function on land should look to the spell air breathing. A caster that wants to drown his foes should look to the spell aqueous orb. A caster that needs a lot of water should look to the spell wall of brine (and get the GM's permission). And a caster that wants to surround himself with weaponized water—and that's a serious trick in Pathfinder as being submerged in water provides total cover from attacks launched from land—should look to the spell seamantle. But if a caster doesn't want to worry about his foes' solid fog and obscuring mist spells—or about the rain soaking his scrolls or death by mundane tsunami—, the spell globe of tranquil water has him covered.
1 Users of sufficient reputation can view my deleted answer.
Note: this answer is based on the interpretation of the spell as projecting a continuous sphere of water, rather than a hollow shell. The text of the spell itself is unclear. The two different interpretations result in, effectively, two very different spells with significantly different overall effects. Thanks to the answer provided by @Chemus for exposing the possibility of the "shell" interpretation.
I don't see why it wouldn't. It even specifically states that it causes the water to act as calm water for the purposes of swim checks - implying that swim checks are appropriate and applicable here. Also, it's a fourth-level blatantly obvious spell that lasts for small numbers of minutes, and requires water-breathing if you don't want to have to hold your breath. Allowing you to swim into the sky at whatever your swim speed is for the duration doesn't seem too overpowered.
I think that your supposition that you can swim through the air is an extrapolation of what the spell says it does, not result of the spell. The spell's description says:
Upon casting this spell, a rippling bubble of calm water extends outward from you to a radius of 20 feet and remains centered on you when you move... All water in this radius counts as calm water for the purposes of Swim checks
There's nothing about being able to swim using the bubble while not otherwise immersed in water, nor about not being able to breathe air. While most bubble-shaped spells are air bubbles, some, like life bubble describe a bubble as a shell around the subject.
I read this as being the same as life bubble; you're surrounded by a shell of water that blocks certain things (any form of moving water, fog or mist, natural or magical, but not temperature or lightning) and keeps water calm within its confines. It goes so far as to hedge out water elementals in the same way that antiplant shell does. Apparently up to and including collapsing if forced against said elemental.
In short, the spell does what it says it does, forming a shell of water that prevents any moving water from affecting the protected region.
