Earth has normal modes of oscillation, many of which are in the milli-Hertz frequency range. Large earthquakes occasionally excite these modes strongly, but the quakes are usually widely spaced in time compared to the ringdown time of a particular mode (typically a few days). There is evidence of a background level of continuous excitation of these modes, with an rms ground acceleration per mode of ∼10⁻¹⁰ cm s⁻² at seismically “quiet” times. The excitation mechanism is suspected to be stochastic forcing by the pressure fluctuations associated with atmospheric turbulence. This exercise deals with some aspects of this hypothesis.

(a) Estimate the rms pressure fluctuations P(f) at frequency f, in a band width equal to frequency △f = f, produced on Earth’s surface by atmospheric turbulence, assuming a Kolmogorov spectrum for the turbulent velocities and energy. Make your estimate using two methods: 

(i) Using dimensional analysis (what quantity can you construct from the energy cascade rate q, atmospheric density ρ, and frequency f that has dimensions of pressure?)

(ii) Using the kinds of arguments about eddy sizes and speeds developed.

(b) Your answer using method (i) in part (a) should scale with frequency as P(f) ∝ 1/f. In actuality, the measured pressure spectra have a scaling law more nearly like P(f) ∝ 1/f^2/3, not P(f) ∝ 1/f. Explain this discrepancy [i.e., what is wrong with the argument in method (i) and how can you correct it to give P(f) ∝ 1/f^2/3?].

(c) The low-frequency cutoff for the P(f) ∝ 1/f^2/3 pressure spectrum is about 0.5mHz, and at 1mHz, P(f) has the value P(f = 1mHz) ∼ 0.3 Pa, which is about 3× 10⁻⁶ of atmospheric pressure. Assuming that 0.5 mHz corresponds to the largest eddies, which have a length scale of a few kilometers (a little less than the scale height of the atmosphere), derive an estimate for the eddies’ turbulent viscosity ν_t in the lower atmosphere. By how many orders of magnitude does this exceed the molecular viscosity? What fraction of the Sun’s energy input to Earth (∼10⁶ erg cm⁻² s⁻¹) goes into maintaining this turbulence (assumed to be distributed over the lowermost 10 km of the atmosphere)?

(d) At f = 1mHz, what is the characteristic spatial scale (wavelength) of the relevant normal modes of Earth? What is the characteristic spatial scale (eddy size) of the atmospheric pressure fluctuations at this same frequency, assuming isotropic turbulence? Suggest a plausible estimate for the rms amplitude of the pressure fluctuation averaged over a surface area equal to one square wavelength of Earth’s normal modes. (You must keep in mind the random spatially and temporally fluctuating character of the turbulence.)

(e) Challenge: Using your answer from part (d) and a characteristic shear and bulk modulus for Earth’s deformations of K ∼ μ ∼ 10¹² dyne cm⁻², comment on how the observed rms normal-mode acceleration (10⁻¹⁰ cm s⁻²) compares with that expected from stochastic forcing due to atmospheric turbulence.