The equation for a traveling wave on the boundary of a two‐dimensional droplet of an ideal fluid is derived by using the conformal variables technique for free surface waves. The free surface is subject only to the force of surface tension and the fluid flow is assumed to be potential. We use the canonical Hamiltonian variables discovered and map the lower complex plane to the interior of a fluid droplet conformally. The equations in this form have been originally discovered for infinitely deep water and later adapted to a bounded fluid domain.The new class of solutions satisfies a pseudodifferential equation similar to the Babenko equation for the Stokes wave. We illustrate with numerical solutions of the time‐dependent equations and observe the linear limit of traveling waves in this geometry.
Satellite observations reveal short pulses in the second time derivative of the geomagnetic field. We seek to interpret these signals using complex empirical orthogonal functions (CEOFs). This methodology decomposes the signal into traveling waves, permitting estimates for the period, angular wave number, and phase velocity. We recover CEOFs from the CHAOS‐6 model, focusing on three geographic regions with strong secular acceleration. Two regions are confined to the equator, while the third is located under Alaska. We find evidence for both eastward and westward traveling waves with periods between 7 and 20 years. There is also evidence for weaker standing waves with complex spatial patterns. Two of the three regions have waves that are compatible with predictions for waves in a stratified fluid. Our results yield estimates for the structure of fluid stratification at the top of the core.
more » « less- Award ID(s):
- 1915807
- NSF-PAR ID:
- 10444589
- Publisher / Repository:
- DOI PREFIX: 10.1029
- Date Published:
- Journal Name:
- Geophysical Research Letters
- Volume:
- 47
- Issue:
- 17
- ISSN:
- 0094-8276
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Abstract -
Recent satellite missions have detected short pulses of magnetic secular acceleration in the equatorial region of Earth’s core (Chulliat et al., 2010; Finlay et al., 2016). The new data provide an opportunity to detect dynamics in the Earth’s core on short timescales. To interpret these signals, we require a technique to separate distinct wave motions. The standard method, called Empirical Orthogonal Function (EOF), applies only to standing waves. An extension to deal with traveling waves (known as complex - EOF) relies on a Hilbert transform of the dataset before applying the EOF methodology (Horel, 1984; Susalito, 1994). This technique allows us to extract the period (T), the angular wave number (m) and the phase velocity (v), based solely on information in the CHAOS-6 model. We focused on two equatorial regions; one centered on Southeast Asia and the other on the Caribbean. The first two complex - EOFs in both regions account for over 90% of the signal. We find two eastward traveling waves in the Southeast Asia region (Tmode1=16.2 years, Tmode2=9.1 years, vmode1 = 3.5 ± 0.7 degrees/year, vmode2 = 7.1± 1.8 degrees/year and mmode1=mmode2=6). In the Caribbean region, the first mode represents a westward traveling wave (Tmode1 =6.7 years, vmode1 = -7.0 ±0.4 degrees/year and mmode1 = 6). The second mode appears to be a standing wave with a complicated spatial pattern. Extending our analysis beyond ±20º latitude causes a gradual loss of coherence, suggesting that the waves are confined to the equator, consistent with predictions for equatorially trapped MAC waves. In fact, both of the eastward waves in Southeast Asia are compatible with a thin layer of strongly stratified fluid in the outer 28 km of the Earth’s core. Confirmation of this result will require forward models to predict the magnetic secular acceleration expected from equatorially trapped MAC waves. As future work, we propose to use these forward models to reconstruct the CHAOS-6 model in the two equatorial windows.more » « less
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We propose a new two-parameter family of hybrid traveling-standing (TS) water waves in infinite depth that evolve to a spatial translation of their initial condition at a later time. We use the square root of the energy as an amplitude parameter and introduce a traveling parameter that naturally interpolates between pure traveling waves moving in either direction and pure standing waves in one of four natural phase configurations. The problem is formulated as a two-point boundary value problem and a quasi-periodic torus representation is presented that exhibits TS-waves as nonlinear superpositions of counter-propagating traveling waves. We use an overdetermined shooting method to compute nearly 50,000 TS-wave solutions and explore their properties. Examples of waves that periodically form sharp crests with high curvature or dimpled crests with negative curvature are presented. We find that pure traveling waves maximize the magnitude of the horizontal momentum among TS-waves of a given energy. Numerical evidence suggests that the two-parameter family of TS-waves contains many gaps and disconnections where solutions with the given parameters do not exist. Some of these gaps are shown to persist to zero-amplitude in a fourth-order perturbation expansion of the solutions in powers of the amplitude parameter. Analytic formulas for the coefficients of this perturbation expansion are identified using Chebyshev interpolation of solutions computed in quadruple-precision.
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