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How is the management fee calculation for a fund with several closings?

How is the management fee calculation for a fund with several closings?
2 See in Base
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With the caveat that everything always depends on legal agreements, normally, the Management Fee is calculated annually on the total fund size that is closed, and fee often starts off larger for the active investment period and reduces over time to an average over the fund life, which is classically 2%.
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A follow up question:
Assuming a $10MM fund with a first close of $2.5MM and a management fee of 2%, the fees start at $50K per year.
Then, in the second close after 3 months, the fund is at $5MM. Will the fees be paid retroactively for the past 3 months as if the first close was at $5MM?
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Each investor will be charged retroactively on the date they are admitted into the fund, as if they were part of the first close.

If the first close is $2.5MM with a management fee of 2%, the fees will start at $50k per year or $12.5k quarterly. 

If a second close takes place 3 months into the life of the fund, bringing the new fund size to $5MM, the LPs that are part of this second close would be charged a catch up fee for the first 3 months of the fund, on the day they are admitted. This would bring the management fees to $100k per year or $25k quarterly. 

In the end, each investor will have a 2% management fee charged to their Capital Account balance in the first year of the funds life. 

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