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I'm working through Vallis, and in the context of talking about a parameterization of friction in a fluid layer using a viscous term, he says:

In general, stress is a tensor $\tau_{ij}$ with an associated force given by $F_i = \frac{\partial \tau_{ij}}{\partial x_j}$, summing over the repeated index. It is common in geophysical fluid dynamics that the vertical derivative dominates, and in this case the force is $\mathbf{F} = \frac{\partial \tau}{\partial z}$. We still use the word stress for $\mathbf{\tau}$, but it now refers to a vector whose derivative in a particular direction (z in this case) is the force on a fluid.

My question is what does $\frac{ \partial \tau}{\partial z}$ mean in this context? In what sense does $\tau$ now refer to a vector? My naïve guess is that it means $\mathbf{F} = (\frac{\partial \tau_{13}}{\partial z}, \frac{\partial \tau_{23}}{\partial z}, \frac{\partial \tau_{33}}{\partial z})$, but I really don't know. Any insight would be very helpful.

For reference, I have the equivalent of a bachelor's (ish) in Applied Mathematics with a healthy amount of undergrad physics, but I'm not super familiar with the conventions regarding tensors (as opposed to matrices), when summing is assumed necessary (which I know is also context dependent).

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    $\begingroup$ Your guess is correct. $\endgroup$ – Deep Jul 13 '17 at 5:26
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Generally these types of equations are simplifications of a term from the Navier-Stokes equation given by $\nabla \cdot \boldsymbol{\tau}$, where $\boldsymbol{\tau}$ is the viscous stress tensor.

The stress tensor is the off-diagonal terms in the total pressure (or stress) tensor. The divergence of these terms relate to the following analog: $\partial_{k} \ \tau_{i,j}$ is the rate of change along the $\mathbf{k}$-direction of the ith component of the momentum flux transported along the jth direction (or through the $\mathbf{i}$-$\mathbf{k}$ plane). This describes shear flows, where one has momentum transport across a boundary layer orthogonal to the flow direction.

In your example, the fluid flow would be orthogonal/perpendicular to the z-direction (e.g., x-direction) and the force would also be orthogonal/perpendicular the z-direction (e.g., y-direction).

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