Our forefathers had devised numerous prayers to propitiate the deities and pray for rains. These mantras can be chanted in times of drought and in extreme heat wave conditions.
It is purely a matter of faith and belief that drives these prayers. Some of them are listed below:
1) Chanting of Virata Parva from the Mahabharata
One of the the Jagadgurus of Sringeri, Sri Sri Chandrasekara Bharati Maha Swamigal has mentioned that the Virata Parva of the Mahabharata has been encoded with Varuna Beejas and Varuna Mantras. Chanting of these chapters of the Mahabharata would serve as a prayer for abundant rainfall.
The text in Sanskrit can be read here : The Mahabharata in Sanskrit
The book "Guru Krupa Vilasam" mentions an interesting incident in 1925 at Devakottai (near Madurai) by Gurupada Shekaran on the Vana Parva chanting and its effect.
Sri Gurupada Shekaran is present during an discussion between Swamiji and the mutt manager.The manager says "It is better if we move our camp to Madurai. Its more than 5-6 years since it rained here. All the wells are empty. We are bringing water travelling a long distance with the help of elephants. There are a lot of people who have food in our camp. It is difficult to manage without water".
Swamiji says "If I was informed earlier we might have avoided coming here. But having come it is not fair to leave this place disappointing the people. Ask the priest to come with Mahabharata book along with puja items". After the priest came, Swamiji asks him to chant Virata Parva and the priests starts off in a traditional way. Swamiji also starts chanting mantras. Gurupada Sekaran witnesses all what was happening. It was around 1 PM in the noon and the sun was at its peak. At 2 PM, clouds from nowhere hid the sun. Cool breeze starts and its starts raining heavily. So heavy was the rain, except the sound of rain nothing could be heard. It continues to pour and at around 4:30 PM, the manager comes to Swamiji with a sign of relief. Swamiji asks "Is the rain sufficient?" and then asks the priest to light the camphor as a sign of completion. At 5 PM, the priest lights the camphor and almost spontaneously the rain stops.
The next day, he humbly enquires about what happened yesterday. Swamiji says "We did Varuna Japa. The rains came. Sage Vyasa has embedded the varuna mantra into the Virata Parva. The effect the of the mantra creates a subtle effect in the environment"
You can see the full extract of the article from Guru Kripa Vilasam in images(in Tamil)
2) Varuna Mantra Japa - This a very detailed procedure and there are lots of regulations to be followed like eating food without salt till the Japa is completed etc. It is usually performed by qualified priests sitting inside water filled tanks for hours. The malayalam movie "Paithrukam" is based on this concept of Varuna Japa in a drought ridden place.
Amurya upa surye
Yabhirva surya saha
Ta no hinvantvadhvaram
Apo devirupa hvaye
Yatra ghava pibanti na
Sindubhya kartva havi
May waters gathered near the Sun, and those wherewith the Sun is joined, speed forth this sacrifice of ours. I call the Waters, Goddesses, wherein our cattle quench their thirst; oblations to the Streams to be given.
Tat tva yami brahmaa
Ahe amano varueha
Bodhyurusasa ma na
I ask this of thee with my prayer adoring; thy worshipper craves this with his oblation. Varuna, stay thou here and be not angry; steal not our life from us, O thou world-ruler.
3) Chanting of Varuna Sukhta and Parjanya Sukhta - Varuna Sukhta is a powerful mantra from the vedas for invoking the rain deity. Parjanya means rain bearing clouds. Chanting of these 2 mantras should be performed by people well versed in Vedas.
4) Chanting of Azhi Malai Kanna Pasuram from Tiruppavai - This Tamil poem should be chanted in groups in dry lakes, river beds etc. in multiple groups. The poem is given below
aazhi mazhai(k) kaNNaa onRu nee kai karavEl
aazhi uL pukku mugandhu kodu aarthu ERi
oozhi mudhalvan uruvam pOl mey kaRuththu(p)
paazhiy am thOLudai(p) paRpanaaban kaiyil
aazhi pOl minni valamburi pOl ninRu adhirndhu
thaazhaadhE saarnga mudhaiththa sara mazhai pOl
vaazha ulaginil peydhidaay naangaLum
maargazhi neeraada magizhndhElOr embaavaay 4
Meaning: Oh Lord of the rains, who is as majestic as that of the beautiful sea, don't hesitate to dive into the ocean, collect as much water, rise to the sky and assume the colour of the Lord (black). As with the Lord, who has graceful and powerful shoulders, oh rain god, without any delay, shine like the discus that he holds in His right hand and resound to produce the same impact that occurs when the conch on his left hand is blown, instantly produce the rain as how the Lord will shoot his arrows, to allow us take bath during the vow and happily flourish and thrive in this world.
5) Purusha Mirugam Puja at Thiruvadavur - Purasha mirugam is known as Spinyx in Egypt. It is commonly used as a vahanam in south Indian temples. It has the head of a Lion and body of a main. Tiruvadavur temple is located near Madurai. Outside the temple there is this statue of the Purusha Mirugam. The locals perform puja to this for abundance in rains. This was followed in several temples in south India but now this practice is lost.
Detailed Article Spinx temple of Thirvathavoor, Near Madurai
6) Chanting of Annapurna Ashtakam and Avahanthi Homam
7) A vedic ritual called Kariri Ishti is described for abundance of rains.
8) "Amruta Varshini" raaga - The name itself means flow of nectar. Songs can be sung in this raaga for rains.
9) Chanting of the Rishya Srunga Sloka - This was suggested by Kanchi Maha Periyava when Tamilnadu was facing drought with no rains.
Sloka in English
Rishyasringaaya munaye vibhandaka sutaayaca
Satyah sadvrishti hetave
Vibhandaka sutah sreemaan
Mahaa varsham prayachathu