Past exam of the physics of atmospheres and oceans course of the University of Oxford / 2017 / 6 / d / Solution

At the specified state, the symmetric matrix isThe planar characteristic equation is ${\lambda }^2+22\lambda -104=0$, giving $4$ and $-26$, while the third eigenvalue is $-6$. An orthonormal set of eigenvectors isWith comparable prior uncertainty in these dimensionless coordinates and equally accurate scalar observations, prioritize the growing combinationAn observation of this projection removes error in the direction that grows most strongly over a sufficiently short forecast; its norm initially grows at rate $2$ in Lorenz time units. The strongly contracting direction is a lower priority under those assumptions. This is not an observation of the absolute state amplitude alone: the relevant error projection is $(\delta x+2\delta y)/\sqrt{5}$. Anisotropic prior covariance, observation noise or a different forecast norm would modify the optimal choice; for a finite forecast the leading right singular vector of the actual propagator replaces this instantaneous direction.