portfolio turnover definition definition, question, starting, shorts, stocks

Siddimaula

What is a precise mathematical definition of portfolio turnover? Here’s a question to tease out the various issues.
Suppose I have a market-neutral fund starting with $1MM. The portfolio goes long a bastket of stocks equal to $1MM and goes short a basket equal to $1MM in value. In 6 months, the long portfolio is replaced with a whole new set of long stocks, and the short portfolio is replaced with a new set of shorts. The longs never overlapped with the shorts. In 1 year, the long and short portfolios are liquidated leaving $1.5 MM in cash.