What is the Boussinesq Term in Momentum Equation

CFDtoy

In the momentum equation, what is the correct representation of the bousinessq body force term?

d(dhoU)/dt+conv term = -gradP + rho*(1-beta*(T-Tref))

or simply

d(dhoU)/dt+conv term = -gradP -beta*(T-Tref))

is the force term = rho*g where rho = rho(T) or simply the difference of actual density rho to reference density rho_0.

Depending on the RHS, the source values are completely off-set. I am not sure if the source values are the differences in density of the actual body force (rho*g) where rho varies as rho = rho_0+rho_0*beta*(deltaT)

In the second case, if its an isothermal situation, does it mean no gravity force is acting on the system?

And, Should, the momentum equation give back simply the rho*g term is no density variation due to temperature changes are available?

Thanks,

CFDtoy