| 1 |
Introduced the Instructor.
Went over the class description and the course syllabus. |
KC01: 1.1-1.7
CR94: Chapter 1 |
Problem Set 1 out |
| 2 |
Gave out a manuscript on dimensional analysis (PDF - 1.0 MB), a manuscript on kinematics (PDF), and some notes on Einstein notation.
Went through a review of some vector stuff, wave kinematics, eigenvectors / values, and dimensional analysis. |
KC01: 3.1-3.5, 3.13
KFF: I, II, and IV |
Problem Set 1 due
Problem Set 2 out
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| 3 |
Discussed scale analysis and gave some examples.
Defined a fluid with respect to its mode of resistance to applied forces.
Discussed the differences between Eulerian and Lagrangian views of the world.
Derived the material derivative. |
KC01: 3.6-3.10 |
Quiz 1 out |
| 4 |
Gave some examples of trajectories, streaklines, and streamlines.
Stated the Cauchy-Stokes decomposition theorem (aka Helmholtz's 1st law): fluid motion can be decomposed into translation, dialation, and rigid rotation.
Derived expressions for linear strain rate, shear strain rate, and rigid rotation rate.
Demonstrated that initial axes can be chosen that will remove all shear strain, leaving only normal strain (dialation) and rigid rotation.
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KC01: 3.6-3.10
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Problem Set 2 due
Returned Quiz 1
Problem Set 3 out
Quiz 2 out |
| 5 |
Introduced the velocity gradient tensor, and explained the terms in the strain rate tensor and the rotation rate tensor.
Provided analytic and numerical examples (G=[4 4;2 8] (ZIP), G=[-1 6;1 -1] (ZIP), G=[-1 6;6 -1] (ZIP), G=[-3 6;-6 -1] (ZIP), these are zipped pdf files. Let me know if you require a different format). (The gexample1.zip file contains: example1five.pdf, example1four.pdf, example1one.pdf, example1six.pdf, example1three.pdf, and example1two.pdf. The gexample2.zip file contains: example2five.pdf, example2four.pdf, example2one.pdf, example2six.pdf, example2three.pdf, and example2two.pdf. The gexample3.zip file contains: example3one.pdf, example3six.pdf, example3two.pdf. The gexample4.zip file contains: example4five.pdf, example4four.pdf, example4one.pdf, example4six.pdf, example4three.pdf, and example4two.pdf.)
Gave example of shear in a rock.
Introduced the concept of a linear uncertainty propagator, and of singular vectors (perpendicular axes at initial time that evolve into the perpendicular axes at final time that have experienced the most stretching or contraction).
Derived how to determine the initial and final time singular vectors using eigenvectors of M'M and MM', respectively. A proof is provided here (PDF). |
KFF: V
KC01: 4.1-4.4 |
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| 6 |
Had another go at describing singular vectors, and gave examples in oceanography and numerical weather prediction.
Introduced the concept of intensive variables (independent of system size [u, T, per volume, per mass]), and extensive variables (depend on size or extent of system [mass, energy, momentum]).
Can use the material derivative to relate Lagrangian and Eulerian rates of change of intensive variables, but need the Reynolds' transport theorem (RTT) to relate Lagrangian and Eulerian rates of change of extensive variables.
Derived various forms of the RTT.
Used the RTT to produce an Eulerian expression for conservation of mass (the continuity equation). |
KC01: 4.5-4.11 |
Returned Problem Set 2
Returned Quiz 2
Problem Set 3 due
Quiz 3 out
Problem Set 4 out |
| 7 |
Discussed the Boussinesq approximation in the context of the continuity equation, and showed that it reduced conservation of mass to a statement of conservation of volume.
Discussed the analastic approximation for the atmosphere.
Started a derivation of the momentum equation beginning with a Lagrangian conservation expression.
Got as far as producing the relevant constitutive relationship (relation between stress and strain). |
KC01: 4.13-4.15 |
Returned Problem Set 3
Returned Quiz 3 |
| 8 |
Explained the physical meaning of the terms in the constitutive relationship.
Plugged the expression for the stress tensor into the momentum equation, and evaluated \Del \dot \tau.
Produced the pressure gradient term, the shear viscosity term, and the divergence viscosity term.
From the resulting Navier-Stokes equations, produced the Euler equations by assuming incompressible and inviscid flow.
Derived the hydrostatic relation, and applied the Boussinesq approximation to Navier-Stokes.
Derived the total energy equation. |
KC01: 4.16-4.17
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Problem Set 4 due
Problem Set 5 out |
| 9 |
Guest lecturer, Greg Lawson.
Derived Bernoulli function form of Euler's equations beginning from the full viscid, compressible Navier-Stokes equations.
Considered three special cases of Bernoulli's form:
- steady flow,
- steady irrotational flow, and
- unsteady irrotational flow.
Considered irrotational flow in general and how zero vorticity allows definition of a velocity potential.
Three examples demonstrate utility and application of Bernoulli's equation. |
KC01: 5.1-5.3, 5.7-5.8, 3.11(review) |
Returned Problem Set 4
Quiz 4 out |
| 10 |
Continued the derivation of the total energy equation.
Derived the mechanical energy equation from the momentum equations, and subtracted it from the total energy equation to obtain the heat equation.
Looked at implications of the Boussinesq approximation.
Introduced potential temperature to produce a heat equation for the atmosphere. |
KC01: 5.1-5.3, 5.7-5.8, 3.11(review) |
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| 11 |
Derived the 2nd law of thermodynamics.
Provided a summary of the closed set of derived equations.
Started a discussion of vortex motion.
Looked at solid body rotation as an example of a rotational vortex, and a point vortex as an example of an irrotational vortex.
Introduced circulation, derived Stoke's Theorem, and showed how the line integral expression for circulation is equivalent to the area integral of vorticity form. |
KC01: 5.4-5.5, and the "Meaning of (\omega \dot \del) u" subsection in 5.6 |
Problem Set 5 due
Problem Set 6 out |
| 12 |
Discussed the choice of sign on the velocity potential and the stream function. Showed that one should try and choose signs that are consistent with the Cauchy-Riemann conditions.
Gave examples of the motion resulting from interacting point vortices, including the use of the method of images near boundaries.
Derived Kelvin's Circulation Theorem via a Lagrangian approach.
Discussed how Kelvin's Circulation Theorem breaks down as various assumptions (inviscid, barotropic, only conservative body forces) are violated. |
KC01: 5.4-5.5, and the "Meaning of (\omega \dot \del) u" subsection in 5.6 |
Quiz 5 out |
| 13 |
Per students' request, a bit of time was spent describing 4th order Runga-Kutta integration.
Introduced the Helmholtz Vortex Theorems, and provided arm waving proofs for each.
Derived the vorticity equation by taking \del cross the momentum equation for both a barotropic and a baroclinic flow.
Described how baroclinicity generates rates of change of vorticity, and how vortex stretching and tilting generates rates of change of vorticity.
Gave a brief midterm review. |
KC01: 4.12
CR94: 2.1-2.5 |
Problem Set 6 due
Problem Set 7 out
Returned Quiz 5 |
| 14 |
Showed how to transform vectors between rotating and inertial frames.
Used the transformation to alter the momentum equations to account for the earth's rotation.
Described that the centrifugal force acts to deform the shape of the earth so that the tangential component of the centrifugal force balances the tangential component of gravity.
Gravity is redefined as the sum of gravity and the centrifugal force.
The coriolis force is introduced and the impact of the coriolis force is demonstrated in a series of movies and demonstrations.
A nice writeup on the Coriolis force can be found here (PDF). |
KC01: 5.6, 14.1-14.4
CR94: 3.1, 3.4-3.5 |
Returned Problem Set 5 |
| 15 |
Introduced the concept of curvature terms necessary in locally cartesian coordinate systems.
Modified the vorticity equation to account for the earth's rotation.
Modified Kelvin's Circulation Theorem to account for the earth's rotation.
Performed some scaling to help simplify the momentum equations in a rotating frame. |
KC01: 14.5
CR94: 4.1-4.2, 3.4 |
Problem Set 7 due
Problem Set 8 out |
| 16 |
Discussed the necessity of eddy diffusivity and then promptly decided to neglect viscosity all together for the time being.
Wrote down the associated simplified momentum equations and introduced the f-plane and the beta-plane approximations.
Showed how these simplified equations can describe inertial circles if radial velocities and radial pressure gradients are ignored.
Produced the geostrophic balance by neglecting the time derivative of velocity, and then demonstrated that this is valid for small Rossby numbers.
Introduced pressure as a vertical coordinate, and looked at geostrophy in atmospheric analyses.
Derived the gradient wind equations to account for some of the discrepancies between the geostrophic and observed winds. |
KC01: 14.5
CR94: 4.1-4.2, 3.4 |
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| 17 |
Continued exploration of the gradient wind.
Solved the gradient wind equation to produce a solution that is a function of Rossby number and non-dimensional pressure gradient.
Used the solution to explain why there are no cyclonic highs, why the strength of anti-cyclonic highs is limited, why the strengh of both cyclonic and anti-cyclonic lows are not (theoretically) limited, and the conditions for cyclostrophic winds.
Discussed isallobaric wind and explained its relevance to weather prediction.
Introduced the Ekman number and explained why we get cross-isobaric flow when friction is important. |
KC01: 14.6-14.7
CR94: 13.1
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Problem Set 8 due
Problem Set 9 out |
| 18 |
Greg Lawson, guest lecturer.
Considered simplifications of the vorticity equation:
- frictionless, barotropic, geostrophic flow leads to the Taylor-Proudman Theorem, and
- frictionless, geostrophic flow leads to the thermal wind equations. Taylor-Proudman implies columnar flow, i.e., vertical rigidity (z-derivatives of geostrophic velocities = 0).
Thermal wind relates expected z-derivatives of geostrophic velocities to horizontal gradients in density, temperature, or entropy (via equation of state).
Showed four movies of a rotating tank highlighting this behavior and considered plots of atmospheric data.
Next, a very basic intro to fluid frictional boundary layers.
Non-rotating fluids have boundary layer depths that grow with time, whereas rotation (Ro << 1) bounds the boundary layer thickness. |
KC01: 14.6-14.7
CR94: 5.1-5.4, 8.1-8.3 |
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| 19 |
Derived the mass transport in the ocean's surface boundary layer, and found that it is orthogonal to the applied surface stress.
Demonstrated the spiraling nature of the vertical structure of the Ekman layer.
Showed that the structure of the surface forcing leads to convergence or divergence of mass in the Ekman layer, and this results in mass exiting or entering the Ekman layer via Ekman pumping or suction.
The geostrophic response to Ekman pumping, Sverdrup transport, was derived, and the necessary existence of a western boundary current was discussed.
Rossby waves were introduced by considering the impact of a ring of fluid displaced in a sinusoidal pattern. |
KC01: 14.8, 14.13
CR94: 4.3 |
Problem Set 9 due
Problem Set 10 out |
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Waves Primer
An optional class that introduced various forms of wave equations, wavelength, wavenumber, period, frequency, phase speed, and group speed.
Movies were shown to qualitatively demonstrate the impact of different values of wavenumbers and frequency. |
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| 20 |
Described the evolution of a Rossby wave in a fixed depth fluid.
Generated a vorticity equation via Kelvin's circulation theorem, linearized, and obtained a single equation in a single unknown by substituting streamfunction expressions for velocities.
Assumed a wavy solution and produced the associated dispersion relation.
Noted that the phase speed is always westward (relative to the mean zonal flow), and that the group velocity can be either westward or eastward (PDF), (PDF), (PDF).
Derived the shallow water equations, and through cross differentiation (PDF), produced the shallow water potential vorticity equation. |
KC01: 14.15
CR94: 4.4, 6.4 |
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| 21 |
Class voted that it would prefer a "final exam" this is actually a second midterm that covers only the material in the rotation part of the course.
Qualitatively discussed shallow water potential vorticity and its implications.
Used the shallow water potential vorticity equation to quantify the behavior of the associated Rossby waves.
Because the shallow water equations describe 3d flow, cannot use streamfunction as in the fixed depth case. Instead, assume that to first order the velocities are geostrophic.
Produced the quasi-geostrophic form of the linearized shallow water vorticity equation, assumed a wavy solution, and obtained the associated dispersion relation, phase speed, and group velocity (PDF), (PDF).
Introduced the Rossby radius of deformation, and noted that it set a scale on the dispersion relation that was missing from the fixed depth Rossby wave dispersion relation.
Derived the dispersion relation for shallow water gravity waves in the absence of rotation. |
KC01: 14.11-14.12
CR94: 6.2-6.3 |
Returned Problem Set 9
Problem Set 10 due
Problem Set 11 out |
| 22 |
Derived the dispersion relation for Poincare waves starting from the shallow water equations, linearizing, and assuming wavy solution.
Demonstrated that in various limits the Poincare waves behave like inertial oscillations or like gravity waves (quantified via the Rossby radius of deformation).
Derived the dispersion relation for Kelvin waves, and demonstrated that the dispersion relation can be substituted back into the assumed solution to obtain quantitative structural information.
Briefly discussed the role of Rossby and Kelvin waves in the delayed-action oscillator model of ENSO.
Gave a brief review (PDF) of the second half of the course. |
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Returned Problem Set 10 |