Mash the tamarind using your hands along with the warm water and strain this liquid. You will have a thick tamarind extract.
Add 1 cup of water to this and set aside.
Heat a pan with 1 teaspoon sesame oil.
Roast the toor dal, coriander seeds, fenugreek seeds, mustard seeds and dry red chilies separately in the same pan beginning with the toor dal and continuing in that order.
Remove from pan and Set aside to cool.
In the same pan, heat the remaining oil and add the chopped lime.
Saute till limes turn a light brown.
Add the tamarind extract, salt, turmeric powder and jaggery and simmer till raw smell of the tamarind goes away.This takes about 6 to 7 mins.
While this is simmering, grind the roasted spices to a smooth powder in a blender and set aside.
Once the raw smell of the tamarind goes away, bring down the flame to the lowest and add the ground powder.
Mix well using a whisk.
Continue to simmer for about 5 mins until oil begins to leave the sides.
Heat 1.5 tablespoon of sesame oil in small pan.
Pop the mustard seeds and add urad dal, curry leaves, green chili and hing.
Once the dal turns a light brown, add the tempering to the kuzhambu and mix well.
Serve hot with rice and any dry curry of your choice.