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One can distribute the divergence term on right hand side and use this definition of material derivative: ¯ ¯ + = ( ′ ′ ) This equation looks again like the Lagrangian equation that we started with, with the same caveats (i) and (ii) as in Eulerian case, and the definition of the mean-flow quantity also for the derivative operator. The ...
More precisely, the divergence theorem states that the surface integral of a vector field over a closed surface, which is called the "flux" through the surface, is equal to the volume integral of the divergence over the region enclosed by the surface. Intuitively, it states that "the sum of all sources of the field in a region (with sinks ...
Mass continuity would require a vertical transport of air along the cold front where there is divergence (lowered pressure). Although this circulation is described by a series of processes, they are actually occurring at the same time, observable along the front as a thermally direct circulation.
Assuming conservation of mass, with the known properties of divergence and gradient we can use the mass continuity equation, which represents the mass per unit volume of a homogenous fluid with respect to space and time (i.e., material derivative) of any finite volume (V) to represent the change of velocity in fluid media ...
The convection–diffusion equation can be derived in a straightforward way [4] from the continuity equation, which states that the rate of change for a scalar quantity in a differential control volume is given by flow and diffusion into and out of that part of the system along with any generation or consumption inside the control volume: + =, where j is the total flux and R is a net ...
A change in the density over time would imply that the fluid had either compressed or expanded (or that the mass contained in our constant volume, dV, had changed), which we have prohibited. We must then require that the material derivative of the density vanishes, and equivalently (for non-zero density) so must the divergence of the flow velocity:
Ekman transport has significant impacts on the biogeochemical properties of the world's oceans. This is because it leads to upwelling (Ekman suction) and downwelling (Ekman pumping) in order to obey mass conservation laws. Mass conservation, in reference to Ekman transfer, requires that any water displaced within an area must be replenished.
The rapid relative change in the coriolis parameter (a function of latitude) near the equator combined with the ITCZ being located north of the equator leads to similar rapid changes in the surface Ekman transport of the ocean and areas of convergence and divergence in the oceanic mixed layer. Using the larger Pacific basin as an example, the ...