We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
A sphere sinking through a chemical gradient drags fluid with it, deforming the gradient. The sphere leaves a trail of gradient enhancement that persists longer than the velocity disturbance in the Reynolds 
$10^{-2}\leqslant Re\leqslant 10^{2}$, Froude 
$10^{-1}\leqslant Fr\leqslant 10^{3}$ and Péclet 
$10^{2}<Pe\leqslant 10^{6}$ regime considered here. We quantify the enhancement of the gradient and the diffusive flux in the trail of disturbed chemical left by the passing sphere using a combination of numerical simulations and scaling analyses. When 
$Fr$ is large and buoyancy forces are negligible, dragged isosurfaces of chemical form a boundary layer of thickness 
$\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D70C}}$ around the sphere with diameter 
$l$. We derive the scaling 
$\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D70C}}/l\sim \mathit{Pe}^{-1/3}$ from the balance of advection and diffusion in the chemical boundary layer. The sphere displaces a single isosurface of chemical a maximum distance 
$\mathit{L}_{Def}$ that increases as 
$\mathit{L}_{Def}/l\sim l/\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D70C}}\sim \mathit{Pe}^{1/3}$. Increased flux through the chemical boundary layer moving with the sphere is described by a Sherwood number, 
$Sh\sim l/\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D70C}}\sim \mathit{Pe}^{1/3}$. The gradient enhancement trail extends much farther than 
$\mathit{L}_{Def}$ as displaced isosurfaces slowly return to their original positions through diffusion. In the reference frame of a chemical isosurface moving past the sphere, a new quantity describing the Lagrangian flux is found to scale as 
$\mathit{M}\sim (\mathit{L}_{Def}/l)^{2}\sim \mathit{Pe}^{2/3}$. The greater 
$\mathit{Pe}$ dependence of 
$\mathit{M}$ versus 
$Sh$ demonstrates the importance of the deformation trail for determining the total flux of chemical in the system. For 
$\mathit{Fr}\geqslant 10$, buoyancy forces are weak compared to the motion of the sphere and the preceding results are retained. Below 
$\mathit{Fr}=10$, an additional Froude dependence is found and 
$l/\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D70C}}\sim Sh\sim Re^{1/6}Fr^{-1/6}Pe^{1/3}$. Buoyancy forces suppress gradient deformation downstream, resulting in 
$\mathit{L}_{Def}/l\sim Re^{-1/3}Fr^{1/3}Pe^{1/3}$ and 
$\mathit{M}\sim Re^{-1/3}Fr^{1/3}Pe^{2/3}$. The productivity of marine plankton – and therefore global carbon and oxygen cycles – depends on the availability of microscale gradients of chemicals. Because most plankton exist in the fluids regime under consideration, this work describes a new mechanism by which sinking particles and plankton can stir weak ambient chemical gradients a distance 
$\mathit{L}_{Def}$ and increase chemical flux in the trail by a factor of 
$\mathit{M}$.
The propagation of a train of mode-2 internal solitary-like waves (ISWs) over a uniformly sloping, solid topographic boundary, has been studied by means of a combined laboratory and numerical investigation. The waves are generated by a lock-release method. Features of their shoaling include (i) formation of an oscillatory tail, (ii) degeneration of the wave form, (iii) wave run up, (iv) boundary layer separation, (v) vortex formation and re-suspension at the bed and (vi) a reflected wave signal. Slope steepness, 
$s$, is defined to be the height of the slope divided by the slope base length. In shallow slope cases (
$s\leqslant 0.07$), the wave form is destroyed by the shoaling process; the leading mode-2 ISW degenerates into a train of mode-1 waves of elevation and little boundary layer activity is seen. For steeper slopes (
$s\geqslant 0.13$), boundary layer separation, vortex formation and re-suspension at the bed are observed. The boundary layer dynamics is shown (numerically) to be dependent on the Reynolds number of the flow. A reflected mode-2 wave signal and wave run up are seen for slopes of steepness 
$s\geqslant 0.20$. The wave run up distance is shown to be proportional to the length scale 
$ac^{2}/g^{\prime }h_{2}\sin \unicode[STIX]{x1D703}$ where 
$a,c,g^{\prime },h_{2}$ and 
$\unicode[STIX]{x1D703}$ are wave amplitude, wave speed, reduced gravity, pycnocline thickness and slope angle respectively.
