= Adjoint equations for Boussinesq scalar mixing
For a smooth incompressible direct trajectory with total <velocity field> $\mathbf U$ and <active scalar> $\Theta$, the negative-constraint <Lagrangian function in constrained optimization> convention gives
$$
\partial_t\mathbf u^\dagger+\mathbf U\cdot\nabla\mathbf u^\dagger-(\nabla\mathbf U)^T\mathbf u^\dagger+\nabla p^\dagger+\nu\Delta\mathbf u^\dagger-\theta^\dagger\nabla\Theta=0,\qquad \nabla\cdot\mathbf u^\dagger=0,
$$
$$
\partial_t\theta^\dagger+\mathbf U\cdot\nabla\theta^\dagger+\kappa\Delta\theta^\dagger-\mathrm{Ri}_B u_y^\dagger=0.
$$
The transpose term is the <formal adjoint> of $\delta\mathbf u\cdot\nabla\mathbf U$; the two couplings transpose <scalar transport> by <advection> and <buoyancy>. For terminal cost $\int\Theta(T)^2$, the terminal data are $\mathbf u^\dagger(T)=0$, $\theta^\dagger(T)=2\Theta(T)$. They are integrated backward along the stored direct trajectory. <No-slip boundary conditions> and <Dirichlet boundary conditions> for the adjoint <velocity> and homogeneous <Neumann boundary conditions> for the adjoint scalar remove the spatial boundary terms.
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