Abstract
Semi‐Lagrangian schemes that advect scalars with high‐order accuracy can give first‐order errors when calculating field accelerations in a steady rotational flow or in a steady deformational flow. The error also appears in a control‐volume advection scheme that has unlimited Courant number stability which permits large time steps. The error is attributed to the neglect of the correlation between the advective change of the velocity field and the velocity field doing the advection. In general the field acceleration varies along a Lagrangian path. Considering several points along the flow path we can extend the calculation of field accelerations to higher order accuracy. This is demonstrated analytically for steady flows and in a barotropic model with N‐cycle time integration. The higher order calculation thus incurs an increased computational cost proportional to order of accuracy. Scalar advection does not incur this additional computational complexity and cost. The rational choice of order of accuracy for time stepping schemes is examined for models in which the time step is restricted by stability constraints. Flow fields associated with merging eddies and convection are diagnosed to obtain the relative magnitudes of field and local accelerations. An error‐cost analysis is done by considering advection with different orders of temporal differencing and various resolutions normalized against the spatial scale of the quantity being advected and the temporal scale of the advecting velocity field. A scaling argument indicates that the temporal differencing can reasonably have truncation terms with lower order than the spatial differencing depending upon how ?x and ?t scale relative to the minimum length scales L and timescales T that are to be resolved in the flow.
Published Version
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