Integral theorems for the gradient of a vector field, with a fluid dynamical application

Abstract

The familiar divergence and Kelvin-Stokes theorem are generalized by a tensor-valued identity that relates the volume integral of the gradient of a vector field to the integral over the bounding surface of the outer product of the vector field with the exterior normal. The importance of this long-established yet little-known result is discussed. In flat two-dimensional space, it reduces to a relationship between an integral over an area and that over its bounding curve, combining the 2D divergence and Kelvin-Stokes theorems together with two related theorems involving the strain, as is shown through a decomposition using a suitable tensor basis. A fluid dynamical application to oceanic observations along the trajectory of a moving platform is given. The potential generalization of the generalized identity to curved two-dimensional surfaces is considered and is shown not to hold. Finally, the paper includes a substantial background section on tensor analysis, and presents results in both symbolic notation and index notation in order to emphasize the correspondence between these two notational systems.

Figures (3)

• Figure 1: An illustration of the use of the gradient tensor theorem in observational oceanography. The (a) vorticity $\zeta$, (b) normal strain $\nu$, and (c) shear strain $\sigma$ in a snapshot of a numerical simulation of a quasigeostrophic eddy termed BetaEddyOne early23-zenodo are shown, each normalized by the Coriolis frequency $f\equiv 2\Omega \sin \phi$ where $\Omega$ is the angular rotation rate of the Earth and $\phi$ is the model's central latitude. The eddy is sampled along the triangular tracks, as described in the text, and the colours of the circular disks show the average values of the three quantities within each triangular cell as inferred from the information along its boundary using (\ref{['flowtheorem']}). Note that the divergence vanishes for this quasigeostrophic model. See the Data Accessibility section at the end of the paper for details on model output availability.
• Figure 2: Another view of the information presented in figure \ref{['vortex']}. Here the instantaneous values of the (a) vorticity $\zeta$, (b) normal strain $\nu$, and (c) shear strain $\sigma$ along the sampling tracks are shown, as found from interpolating within the model. These are not directly observable because the gradient can only be computed in the along-track direction. For reference, velocity gradient components locally parallel and perpendicular to the sampling tracks are shown in each panel, indicating the information that can be recovered directly from taking along-track derivatives. Dots show the cell-averaged values, generally providing a good match to the local structure, apart from in panel (b) where the normal strain $\nu$ exhibits small-scale oscillatory behaviour.
• Figure 3: An illustration of the gradient tensor theorem. The upper row shows vector fields $\boldsymbol{u}$ consisting of $\boldsymbol{u}=\mathbf{I}\boldsymbol{x}=\boldsymbol{x}$, $\boldsymbol{u}=\mathbf{J}\boldsymbol{x}$, $\boldsymbol{u}=\mathbf{K}\boldsymbol{x}$, and $\boldsymbol{u}=\mathbf{L}\boldsymbol{x}$, respectively, where $\boldsymbol{x}$ is the position vector with respect to the origin. The heavy grey curve in all plots is the bounding curve $\mathcal{L}$, encompassing a region $\mathcal{A}$ over which we wish to integrate and shown with its exterior normal vector in orange. In subsequent rows, the original vector field is multiplied by $\mathbf{J}^T$, $\mathbf{K}$, and $\mathbf{L}$ respectively; the use of the grey colour for the vectors indicates that this is a modified version of the original field. These multiplications turn each of the four original fields into each of the other types. Along the diagonal, marked by the bold bounding boxes, the original vector field is transformed into a purely divergent field. This illustrates how the gradient tensor theorem can be thought of as the divergence theorem applied to versions of the original vector field modified by the transpose of each of the $\mathbf{I}\mathbf{J}\mathbf{K}\mathbf{L}$ tensors.