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

Expand the vector field about the reference trajectory:Over a sufficiently short interval, the tangent-linear propagator isConsequentlyIf $S=J+J^T$ has $S\mathbf{e}=\lambda \mathbf{e}$ and the perturbation is initially parallel to $\mathbf{e}$, its squared Euclidean norm obeysPositive $\lambda$ means instantaneous norm-squared growth, negative $\lambda$ decay, and zero $\lambda$ no first-order change. The corresponding norm growth rate is $\lambda /2$, not $\lambda$. These are eigenvalues of the symmetric part governing the chosen error metric, not generally the eigenvalues of the nonsymmetric Jacobian matrix or the long-time Lyapunov exponents. The direction and the reference Jacobian matrix evolve during a finite forecast.