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We examine correlations of the Möbius function over $$\mathbb{F}_{q}[t]$$ with linear or quadratic phases, that is, averages of the form 1 $$\begin{eqnarray}\frac{1}{q^{n}}\mathop{\sum }_{\deg f0$$ if $$Q$$ is linear and $$O(q^{-n^{c}})$$ for some absolute constant $c>0$ if $$Q$$ is quadratic. The latter bound may be reduced to $$O(q^{-c^{\prime }n})$$ for some $$c^{\prime }>0$$ when $Q(f)$ is a linear form in the coefficients of $$f^{2}$$ , that is, a Hankel quadratic form, whereas, for general quadratic forms, it relies on a bilinear version of the additive-combinatorial Bogolyubov theorem.
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Focusing deep-water surface gravity wave packets: wave breaking criterion in a simplified model
Geometric, kinematic and dynamic properties of focusing deep-water surface gravity wave packets are examined in a simplified model with the intent of deriving a wave breaking threshold parameter. The model is based on the spatial modified nonlinear Schrödinger equation of Dysthe ( Proc. R. Soc. Lond. A, vol. 369 (1736), 1979, pp. 105–114). The evolution of initially narrow-banded and weakly nonlinear chirped Gaussian wave packets are examined, by means of a trial function and a variational procedure, yielding analytic solutions describing the approximate evolution of the packet width, amplitude, asymmetry and phase during focusing. A model for the maximum free surface gradient, as a function of $$\unicode[STIX]{x1D716}$$ and $$\unicode[STIX]{x1D6E5}$$ , for $$\unicode[STIX]{x1D716}$$ the linear prediction of the maximum slope at focusing and $$\unicode[STIX]{x1D6E5}$$ the non-dimensional packet bandwidth, is proposed and numerically examined, indicating a quasi-self-similarity of these focusing events. The equations of motion for the fully nonlinear potential flow equations are then integrated to further investigate these predictions. It is found that a model of this form can characterize the bulk partitioning of $$\unicode[STIX]{x1D716}-\unicode[STIX]{x1D6E5}$$ phase space, between non-breaking and breaking waves, serving as a breaking criterion. Application of this result to better understanding air–sea interaction processes is discussed.
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- Award ID(s):
- 1634289
- PAR ID:
- 10130719
- Date Published:
- Journal Name:
- Journal of Fluid Mechanics
- Volume:
- 873
- ISSN:
- 0022-1120
- Page Range / eLocation ID:
- 238 to 259
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/jfm.2019.428
