Muspell

Muspell (ムスペル) in the world of Toaru Majutsu no Index refers to the flame army that is led by the giant Surtr to oppose the gods in Norse mythology. Based on this legend, the Muspell spell is introduced in Shinyaku Toaru Majutsu no Index in use by Gremlin (Norse) during their invasion of Tokyo.

Etymology
Muspell is a term from Norse Mythology, which has a debated etymology but is related to the fire which will consume the world at Ragnarok.

Muspellheim, the world of fire, is one of the nine worlds supported by Yggdrasil, and the fire giants which dwell there, led by Surtr, who will set the world ablaze at Ragnarok, are referred to as the sons of Muspell.

Appearance
When the proper conditions are met and the spell activates, sparks appear, and then immediately afterwards, giant pillars of fire appear before its target. The pillars of fire are then twisted to form human figures.

Principles
Muspell is a Norse mythology based spell. According to Index, it uses the symbols of army, enemy, flame, and heat. Muspell is a group of automatic attack spells set to feel artificial hatred for those who match certain conditions, which according to Leivinia Birdway are those who are not "stalled" and they appear when people exceed a certain speed. This refers to the situation in Tokyo during the invasion in which the people are unable to do certain actions due to putting pressure on them, like cutting of escape routes in the city, or harassing them with Níðhöggr, in order to distribute what people stood to gain, a way for  to stop people from uniting in order to stop them.

Muspells can attack by taking in air, which swells them up like round balloons, heat it up to over 1400 degrees Celsius, and then blow it back out with tremendous force. Muspells themselves are around 1400 degrees and can easily kill humans by embracing them or even just breathing on them.

Freyja is able to track if Muspells are "reacting", most likely referring to it being activated, as well as the amount of it being destroyed and where they are located. It is through this that Freyja is able to track down Kamijou Touma.

It is unknown how managed to insert the spell into the entirety of Tokyo and its nearby waters.

Weaknesses and countermeasures
Without the regenerative abilities like that of Stiyl Magnus' Innocentius, Muspell is easily negated by Touma. In fact, the ease of Touma being able to destroy them is what drew Freyja towards him. Moreover, if the target is fast enough the moment the Muspell activates, they can lose sight of their target the instant they materialize. They are apparently also vulnerable to great amounts of water, like the ocean.

Since Norse mythology is a mythology of mutual destruction there are enemies prepared for each of the gods, including that of Surtr and his Muspell. Based on this principle, Muspell can be countered, as shown by both Leivinia Birdway and Lessar. Because of their actions, the Muspell's weakness is apparently related to the wind.

Leivinia Birdway used her wand, transforming it into a sword, representing the element of air according to principle of symbolic weapons, to draw in the blazing wind that Muspell blew out. The blazing wind then spun around her dagger, this then allowed Leivinia to thrust the blazing wind into the waters below her, dissipating it. Leivinia Birdway's anti-Muspell attack chant:

"The color is yellow, the shape is a dagger. With the action of defense and the symbol of flow and change, one of the five great internal symbols of my temple shall be contained in my hand!!"

For Lessar, she used her Steel Glove, a strange polearm she carries, against the Muspell itself. Using the blade fingers of the Steel Glove, she formed a special sign with it and then swung it to the side. With this, the Muspell flame figures before her got ripped apart regardless of the distance, as if a powerful gust of wind blew them away. Lessar's anti-Muspell chant:

"The king of the world of fire and the weather god Freyr both die after battling each other. The deer antler carried by Freyr could not be removed by the followers of the king! The masses who cannot match the king even as a group can only accept defeat!! Let the subjugation of the strict scales fill you with the order of my temple!!"