Chris reviewed the mechanics of the advection-diffusion simulation exercise one last time, with students behaving as bins with their height as temperature.



Then he continued providing context for adding advection to the flow analysis for homework #6 (u in x₂ and x₃)

Advection is different by depth — the u₂ term is a function of x₃: f(x₃)

equation of motion: force on a parcel = mass of the parcel times the acceleration of the parcel:

F = m*a

Force is the sum of surface force (stress) and body force (gravity on complete volume)

We looked at surface forces comprising of pressure (which is uniform in all directions) + deviatoric (normal if perpendicular, sheer if not)

We can describe stress as a rank 2 tensor with magnitude and two directions (direction of the force + the face effected)

Students played with all possible normal and sheer force representations on a paper cube to generate T_i,j as a matrix whereby T_mn is the deviatoric stress of an axis m on a face n (there are nine combinations whereby T₁₁, T₂₂, and T₃₃ are normal). Values can be positive or negative depending on whether the face is in positive or negative coordinates:



To get the total surface stress, we want to integrate over ▽ . n * δS

which can then simplify by revisiting Gauss' divergence theorem which suggests:

integral of q . n * δS = ▽ . q

as a very good approximation where dS is δx₁ δx₂

We can then expand F = ma into components body force + surface force (as integrals)

but the integral of δS is hard to mix with δV so we use Gauss' divergence theorem to simplify via:



whereby the green is the body force and the blue is the surface force

We end up with an advection formula that we can express in model code, such as the one we are using in homework #6:

u₂(x₃) = (0.5*B/μ) Z² + (((U_o/h) - (0.5*B*h/μ))

B is the lateral pressure gradient
h is the height of x₃ from the bottom
Z is an array of depths in x₃
U_o is the river runoff rate
μ is a viscosity parameter