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For a differentiable objective on an admissible real Hilbert space $H$, with $\Vert {\mathbf{u}}_0{\Vert }^2/2=E_0>0$, stationarity on the sphere in a normed vector space of fixed kinetic energy means that the projected gradient is normal to the sphere in a normed vector space. Thus $P_H\nabla J=\lambda {\mathbf{u}}_0$ for a real Lagrange multiplier. It follows by testing all tangent variations orthogonal to ${\mathbf{u}}_0$. The multiplier need not be positive; second-order conditions are still needed for a local minimum.

Use $\nu =R{e}^{-1}>0$, $\kappa =P{e}^{-1}>0$, $\mathbf{U}=\overline{\mathbf{u}}+\mathbf{u}$, and $\Theta =\overline{\theta }+\theta$. Work along a sufficiently regular direct trajectory satisfying the given constraints. The zero spatial mean of the scalar field is conserved by incompressible flow, impermeable walls and zero scalar flux, so minimizing $J=(\Theta (T),\Theta (T))$ is equivalent to minimizing scalar variance, up to the fixed domain volume.

The printed functional fixes the initial state to a candidate ${\mathbf{u}}_0$; it contains no term that enforces its kinetic energy. For the optimization over that candidate, add the real Lagrange multiplier constraintEquivalently, one can restrict all control variations to the sphere in a normed vector space of fixed kinetic energy. This term changes the initial-control optimality condition, not the interior adjoint equations.

Let $\mathbf{v}=\delta \mathbf{u}$, $\sigma =\delta \theta$ and $\pi =\delta p$. Linearization of the momentum and scalar transport residuals givesBoth appearances of the perturbation velocity in the nonlinear momentum term have been differentiated. In particular, the coefficient is the gradient of the total velocity, not just the base shear.

For the negative-constraint convention of the functional, integration by parts gives the interior coefficients of $\mathbf{v},\sigma ,\pi$ asThus the adjoint equations for Boussinesq scalar mixing areThe transpose is essential: the $j$th component of $(\nabla \mathbf{U}{)}^T{\mathbf{u}}^{†}$ is ${\sum }_i{u}_i^{†}{\partial }_j{U}_i$. The coupling $-{\theta }^{†}\nabla \Theta$ transposes advection of the scalar by a velocity perturbation; $-R{i}_B{u}_y^{†}$ transposes buoyancy feedback. Dropping the latter would give a passive scalar adjoint, not the active scalar problem.

These equations are integrated backward, not forward. If $\tau =T-t$, they readwith direct coefficients evaluated at $t=T-\tau$. Both terms from the diffusion equation now have the usual forward sign in $\tau$. A direct-adjoint looping method stores or reconstructs the forward trajectory, solves these equations backward, and uses the initial adjoint as the control gradient. The endpoint and fixed-energy conditions below give necessary conditions for a local optimizer, not a global optimality theorem.

The temporal integration by parts terms areThe terminal states are free and the objective has no explicit terminal-velocity dependence. ThereforeThe factor two follows from the displayed objective without a $1/2$ prefactor. The initial scalar field perturbation is fixed, so $\delta \theta (0)=0$ and there is no independently prescribed adjoint-scalar condition at $t=0$. Its value there is obtained by backward integration. Independent variation of the initial velocity state gives ${\mathbf{u}}^{†}(0)={\mathbf{u}}_0^{†}$.

Variation of the candidate ${\mathbf{u}}_0$ in the energy-augmented functional gives the fixed-energy initial-condition optimality condition. For the usual divergence-free no-slip control space $H$, write $P_H$ for its orthogonal projection, the Leray-Helmholtz projection in the usual incompressible velocity space. This is the standard formal control condition when pressure is determined by incompressible flow and normal momentum. If the independently imposed direct pressure flux is retained as an additional constraint, the admissible control variations must also satisfy its compatibility conditions along the trajectory; they need not fill the usual space $H$. The standard projection formula alone does not prove stationarity for that more restricted problem. ThenFor the natural solenoidal adjoint space, $P_H{\mathbf{u}}^{†}(0)={\mathbf{u}}^{†}(0)$. Equivalently, its component tangent to the sphere in a normed vector space of fixed kinetic energy vanishes:Here $E_0>0$ is needed for the sphere in a normed vector space to be regular and for division by $2E_0$. The multiplier is real and can have either sign; normalizing the initial adjoint with a prescribed positive sign is not a general necessary condition for minimization. If $E_0=0$, the only feasible initial velocity is zero and this tangent formula for the sphere in a normed vector space is inapplicable. First-order stationarity alone also allows maxima or saddles; a local minimum requires the appropriate nonnegative constrained second variation.

The spatial boundary terms vanish with periodic adjoints in $x,z$, homogeneous no-slip boundary conditions ${\mathbf{u}}^{†}=0$ at $y=\pm 1$, and ${\partial }_y{\theta }^{†}=0$ there. To see this, velocity variations vanish at the wall while their normal derivatives need not, so the viscous boundary term forces the adjoint velocity to vanish. Scalar variations have zero normal derivative but free values, forcing the adjoint scalar's normal derivative to vanish. Pressure variation gives adjoint incompressible flow; no independent terminal or initial datum is assigned to $p^{†}$. Its additive time-dependent constant may be fixed by a zero spatial mean.

In particular, a homogeneous Neumann condition for the direct pressure does not by variational transposition require ${\partial }_y{p}^{†}=0$. For a smooth adjoint solution, the wall-normal adjoint momentum equation instead suppliesat the flat walls, because the direct scalar normal derivative is zero there. A zero adjoint-pressure derivative would be an additional compatibility choice, valid only when this right-hand side vanishes.

There are also literal direct-data compatibility qualifications in the printed setup. The initial total scalar field is $\overline{\theta }=-\mathrm{erf}(30y)$, whose wall derivative is $-60e^{-900}/\sqrt{\pi }\ne 0$. It is extraordinarily small but mathematically not zero. Thus exact initial scalar data and exact zero wall flux are incompatible for a classical solution smooth at $t=0$. One may use a parabolic mild solution of an abstract Cauchy problem with the boundary condition enforced for $t>0$, or replace the initial profile by an exactly compatible smooth one; these are conventions, not an equality $e^{-900}=0$. The reference scalar field is also not a stationary profile of the diffusion equation at finite $Pe$: the full $\Delta \Theta$ term must be retained.

The pressure compatibility at a no-slip wall is similarly important. Direct normal momentum gives ${\partial }_{y}p=\nu \Delta u_y-R{i}_B\theta$. The separately imposed zero pressure derivative requires this right side to vanish. It is not automatic for arbitrary fixed-energy initial velocities. For example, the divergence-free no-slip field generated by the periodic stream function $S=A(1-y^2{)}^2\sin (\alpha x)$ with $\alpha =\pi /L_x>0$ has $u_x={\partial }_{y}S$, $u_y=-{\partial }_{x}S$ and $\Delta u_y{|}_{y=\pm 1}=-8A\alpha \cos (\alpha x)\ne 0$, while $\theta (0)=0$. With $A\ne 0$ adjusted to any positive energy, no classical solution smooth up to the initial wall can satisfy that extra pressure condition. The displayed adjoints are the formal necessary equations along admissible smooth trajectories; a well-posed physical formulation normally determines pressure from incompressible flow and normal momentum, rather than imposing independent homogeneous pressure flux for every control.

In a normed vector space, this set consists of all points at a fixed positive norm distance $r$ from $x_0$. It also makes sense in an infinite-dimensional Hilbert space, unlike a specifically Euclidean sphere. In a real Hilbert space, the sphere in a normed vector space $\Vert u{\Vert }^2=2E_0>0$ has tangent variations $v$ satisfying $(u,v)=0$; this follows by differentiating the fixed squared norm. This is the geometric constraint used in fixed-energy initial-condition optimality.