Ornamentation: rolls, cranns, and strike cranns on the ocarina
In a previous article, we learned about cuts and strikes, fingered articulations that allow us to separate notes without using the tongue. They work by sounding a higher pitch (cut), or lower(strike), and notated like this:

Rolls and cranns are ornaments formed from sequences of cuts and strikes. They can serve a number of functions, including creating emphasis, and articulating a string of notes.
Rolls — articulating multiple notes in sequence
Have a go at playing a sequence of notes at the same pitch using either cuts or strikes, using one finger. Start out at a low tempo and gradually speed up.

What you'll find is that at a low tempo this doesn't pose much difficulty, but as you speed it up there will be a limit to the tempo you can achieve. Instead of doing this, we commonly alternate between cuts and strikes because it enables the work to be spread across two hands.
The following shows how this would be played. Tongue the first note and play the remaining 2 in a single slur. Articulate the second note using a cut and the third by using a strike. Spend a few minutes playing through this repeatedly and particularly pay attention to getting both the cut and strike exactly on the beat.
As fingered articulations are only effective when the instrument is sounding, it is normal to slur into this pattern from a previous note, as done here.

In Irish traditional music, a single instance of the pattern is called a 'roll'. In that idiom it is considered an ornament, but in reality its essentially both an ornament, and an articulation at the same time. It produces two notes articulated in an ornamental way.
Also while these are most common in Irish music and similar traditions, there's no reason they can't also be applied to other genres. What I've shared in this article is intended to generalise the concept.
Practising rolls
Rolls can be performed on every note on the ocarina besides the highest, lowest, and subholes. Most accidentals can also be rolled. The only requirement is to have a finger covering a hole and a free raised finger to strike with. Preferably, these should be on two different hands.
I'd recommend practising a roll on all of the 'rollable' notes of the C major scale:

Practice each one of these in sequence. If you're a beginner you might want to learn one or two of them a day.
The ocarina offers more freedom than tubular instruments because strikes can be played with any free raised finger against its respective open hole. To give some ideas of how these could be performed:
- Both low E and F can be rolled by cutting with the left middle finger and striking with the right ring finger.
- G and A can be rolled by cutting with the left middle finger and striking with the right index finger.
- High B and C can be rolled by cutting with the left thumb and striking with the right index finger.
- High D and E can be rolled by cutting with the left pinky and striking with the right index finger.
However, note that these fingerings are just a suggestion, and its best to use whatever fingering works best for you.
Once you've got the hang of playing a few of these it would be a good idea to try applying them in some music. Either make up some simple melodies, or find some songs or tunes that have repeated notes, and try articulating them with a roll.
The cross fingered accidentals can also be rolled using the same ideas, but it is not advisable to attempt to 'strike' a subhole or part-covered split hole. The brief duration of a strike is created by a finger bouncing off the instrument. Achieving the same thing with a finger slide is practically impossible.
More examples
Being essentially an articulation of sequential notes, rolls can be applied to notes of any duration, they do not have to be equal.

They work especially well in articulating 3 8th notes of the same pitch, particularly in compound time signatures like 3/8, 6/8, and 9/8. The first example with 3 8th notes of the same pitch is called a long roll in Irish traditional music.

It is also possible for the cut and struck note to be of different pitches. This only really works if the struck note is lower as strikes aren't very effective on ascending intervals.

Rolls can also be performed without being slurred into from a preceding note, in which case the first cut will be a tongued cut. The tongue lowers to start the note at the same time as the cutting finger is lifted.

If you continue to alternate cuts and strikes, you can separate an arbitrary number of sequential notes. Below is the pattern used to articulate a string of 8th notes.

It can be encountered often in tunes as well, for instance in the beginning of Jim Ward's Jig, with 4 notes in sequence. It is common to separate them by tonguing the first, cutting the second, striking the third, and cutting the fourth. This is essentially a roll and a half.

The term 'roll' is often defined strictly as an articulation of 3 sequential notes, and within that definition the above patterns are not technically 'rolls'. However to me that distinction is kind of a pointless.
We often use alternating cuts and strikes because it's easier, and if every possible form had its own name, you would quickly end up with a huge number of names for things that are essentially the same.
Rolls: as emphasis
One of the advantages of fingered articulations is that they respond exceptionally quickly. We can play several of them very close together in time to create a kind of 'drum tap' or double articulation.
Say you have two half notes in series and you want to emphasize the second. You can use a roll with tight timing to do this as shown in the following notation. Note that right now we are just introducing the concept, it does not matter if you can play these right now.

Going further still, 'double cut' and double strike rolls exist. They are performed halving the duration of the first note and inserting a second cut. Performing these is best done by cutting with multiple fingers. How to do so is discussed in 'cranns' later.

It is also worth noting that a roll doesn't need to have a very short duration to be emphatic in all cases. If you have a very long note, even a fairly widely spaced roll can be effective. You should note that the emphatic effect depends on context. For example, if you have a sparse melody then have a sequence of rapid articulations at the beginning of a note, it stands out. If the melody is already busy, however, the effect would be less pronounced and may go unnoticed. It may be more effective to insert an unexpected long note in the latter case.

Learning how to perform these kinds of rapid patterns is best done by first performing them very slowly, and gradually working up the tempo.
Cranns
On the ocarina, there are a few notes which are impossible to roll. Assuming C fingerings, low C and any subhole notes are impossible to strike as there are no raised fingers. The highest note cannot be cut as there are no covered holes. It is still possible to achieve a similar effect on these notes, which is commonly called a crann.
The basic idea of a crann is to separate multiple notes in sequence using only cuts. As playing multiple sequential cuts on the same finger at speed is very difficult, multiple fingers are used. One finger can prepare to cut while the other is still in the air.
The next example shows a general use of a crann. To perform this one the cuts may be performed with the left middle finger and the second using the right index finger. Make sure that the first cutting finger has returned to its hole before performing the second cut. This is shown in the first demonstration in the video.

Cranns are most often used on notes that are impossible to strike, although it is possible to use them on any note where enough fingers are covering holes to play the required cuts.
If you wish to separate multiple notes in sequence, you can loop through the fingerings. To articulate 4 C's in sequence, cut in the order of: left middle finger, right index finger, left middle finger, right index finger. Doing this generally gives enough time for the finger to recover from its prior action.
Cranns can also be used to articulate longer sequences of notes, and notes with differing rhythmic values as in the second example. In the case of a straight rhythm, for instance 4 8th notes.
This can be done by continuing to alternate the left middle finger and right index finger. In the case of 3 sequential notes as in the second example, I find it easiest to perform the cuts using 3 different fingers—here, the left middle, left ring, and right index finger.

If the rhythmic value of the notes is shortened, the crann can also be used in an emphatic form. The 8th notes would be replaced with 16th or 32nd notes. This is basically the same as an emphatic roll, but done using only cuts.
Strike cranns
A strike crann takes the same idea as a crann but uses only strikes, and can be used on the ocarina's high notes.
As far as I know, strike cranns are unique to the ocarina. Rolls and cranns may be performed on any tubular wind instrument but, on such instruments, it is only possible to play a strike using the finger below the lowest closed hole.
The fact that an ocarina's entire chamber oscillates is what makes them uniquely suited to strike cranns, as strikes can be played using multiple fingers and any open hole.
For instance, the first example here can be played by striking with the left middle finger and right index finger. In the second emphatic example a 3rd strike may be performed on the left middle finger. Like the demonstration of the emphatic crann it is played slowly.

Notating rolls, cranns, and strike cranns
In my opinion, the best way to notate rolls, cranns, and strike cranns for ocarina music is to notate the individual cuts and strikes, over notes describing the intended rhythm.
It makes sense to name more specific variations of these patterns where they are used within a single idiom. For example, Irish music has 'long' and 'short' rolls. The following example shows two different notations for an Irish long roll, and how it is played to the right. A roll can be noted by an arched line or a tilde (~) above a note.

It works in such a context because the number of forms used is quite limited, but were names like this to be applied to ocarina music broadly, considering the instrument is played by people from countless backgrounds, this would quickly result in a huge number of names.
I feel this would be confusing as they would be names for slight variations of the same thing. The specifics would be closely tied to the intended effect in a given piece of music and will have questionable applicability in a general sense.
If you are interested in learning more about the uses of this ornamentation in Irish traditional music, I recommend reading The Essential Guide to Irish Flute And Tin Whistle. It is exhaustively detailed and is easy to adapt to the ocarina.