"There is heavy rain."
Translation:هُناك مَطَر ثَقيل.
First of all, it is مطر = rain, not مطار = airport.
As for your question, it depends on the meaning. The sentence says "There is a heavy rain" .. In this formulation, the word "rain" should be indefinite and no one says "there is the heavy rain" !! The same applies in Arabic.
Grammatically, This is a nominal sentence. Nominal sentences have two parts: The "beginner المبتدأ", it is a definite word. and the completion الخبر, which completes the information and is usually an indefinite word (could be a word with a preposition, or other cases).
Therefore, in the example above, the pronoun "هناك" is definite (all pronouns are), so the word "مطر" should be indefinite. The word "ثقيل" is an adjective to "مطر".
Actually the last bit of your explanation is incorrect. In this sentence هناك is الخبر. Because it is placed at the beginning of the sentence, we call this الخبر المقدَّم. So مطر ثقيل is المبتدأ, which is called المبتدأ المؤخر because it is placed at the end of the sentence. The reason why the sentence doesn't start with المبتدأ is that an Arabic sentence cannot have an indefinite noun at the beginning. It's the same in sentences like عندي بيت or لي أخت. Here also the sentence starts with الخبر and ends with المبتدأ.