University:
Widener University
Course:
ESSC 103 | Intriduction To Earth Science
Academic year:

2023
Views:

161

Pages:

2
Author:

Agustin Santana

Barotropic vorticity lab Equations: We can solve the equations for homogeneous, incompressible, two–dimensional ﬂow p d u + f k̂ × u = −∇ + ν∇2 u dt ρ ∇·u=0 using a streamfunction u = −∇ × k̂ψ(x, y, t) or u=− ∂ ∂ ψ, v= ψ ∂y ∂x From the divergence of the momentum equations, we ﬁnd ∇2 ∂uj ∂ui p = ∇ · f ∇ψ − ρ ∂xi ∂xj so, given ψ, we can ﬁnd the velocities and the pressure p/ρ. From the curl of the momentum equations, we ﬁnd the vorticity equation d q = ν∇2 q dt with ∂ ∂ v− u = f + ∇2 ψ ∂x ∂y Given ψ, we can ﬁnd q and then evaluate the advection and diﬀusion terms to step q forward in time. Inverting the Laplace operator allows us to calculate ψ at the new time. The domain is doubly periodic — ﬂow out the left comes back in the right and ﬂow out the bottom comes back in the top. ˆ·∇×u=f + q =f +k Problems: The model allows you to specify the initial ψ(x, y, 0) ﬁeld and then it calculates the other ﬁelds and displays them. You can also watch the advancement in time of the streamfunction. Problem #1: Consider a single vortex 1 1 � x �2 1 � y �2 − ] ψ = U0 L exp[ − 2 2 L 2 L (or equivalently) ψ = U0 L∗gauss(x/L, y/L). How do ∆ψ and ∆(p/ρ) compare for diﬀerent U0 , L values? In a geostrophic state, these would be related by ψ = f ρp . Calculate the pressure analytically using polar coordinates. How big are the ageostrophic terms and how does the sign of the streamfunction aﬀect the relationship? How does this evolve when β = df /dy is not zero? 1 Problem #2: What happens to elliptical vortices? Discuss the ﬂow patterns and the evolution. Problem #3: Consider two vortices with (a) the same sign or (b) opposite signs. Describe the behavior for diﬀerent separations. Problem #4: Consider a single vortex in a shear ﬂow ψ = U0 L ∗ gauss(x/L, y/L) + Us W cos(πy/W ) What happens for diﬀerent strengths and signs of the vortex/ shear? You can also try diﬀerent y locations and sizes. Note — the shear ﬁeld must be periodic over the northsouth extent of the domain. Recreating runs You can save snapshots from a time-evolution calculation by setting the rate on the page which controls the time-evolution. The data is then in the /tmp/mocha-lab-$USER directory as qg1.in (the input data), fort.1 (the ψ ﬁelds) and fort.2 (the q ﬁeld). You can copy these into your own directory for replaying or just the input data for rerunning. See analyze.m for ways to access the data. 2