Young, rapidly rotating M dwarfs exhibit prominent starspots, which create quasiperiodic signals in their photometric and Doppler spectroscopic measurements. The periodic Doppler signals can mimic radial velocity (RV) changes expected from orbiting exoplanets. Exoplanets can be distinguished from activity-induced false positives by the chromaticity and long-term incoherence of starspot signals, but these qualities are poorly constrained for fully convective M stars. Coherent photometric starspot signals on M dwarfs may persist for hundreds of rotations, and the wavelength dependence of starspot RV signals may not be consistent between stars due to differences in their magnetic fields and active regions. We obtained precise multiwavelength RVs of four rapidly rotating M dwarfs (AD Leo, G227-22, GJ 1245B, GJ 3959) using the near-infrared (NIR) Habitable-zone Planet Finder and the optical Keck/HIRES spectrometer. Our RVs are complemented by photometry from Kepler, TESS, and the Las Cumbres Observatory network of telescopes. We found that all four stars exhibit large spotinduced Doppler signals at their rotation periods, and investigated the longevity and optical-to-NIR chromaticity for these signals. The phase curves remain coherent much longer than is typical for Sunlike stars. Their chromaticity varies, and one star (GJ 3959) exhibits optical and NIR RV modulation consistent in both phase and amplitude. In general, though, we find that the NIR amplitudes are lower than their optical counterparts. We conclude that starspot modulation for rapidly rotating M stars frequently remains coherent for hundreds of stellar rotations and gives rise to Doppler signals that, due to this coherence, may be mistaken for exoplanets.

A three-dimensional, steady, laminar shear-layer flow spatially developing under a boundary-layer approximation with mixing, chemical reaction, and imposed normal strain is analyzed. The purpose of the study is to determine conditions by which certain stretched vortex layers appearing in turbulent combustion are the asymptotic result of a spatially developing shear flow with imposed compressive strain. The imposed strain creates a counterflow that stretches the vorticity in the span wise direction. The equations are reduced to a two-dimensional form for three velocity components. The non-reactive and reactive cases of the two-dimensional form of the governing equations are solved numerically, with consideration of several parameter inputs such as Damk ̈ohler number, Prandtl number, chemical composition, and free-stream velocity ratios. The analysis of the non-reactive case focuses on the mixing between hotter gaseous oxygen and cooler gaseous propane.The free-stream strain rate κ∗ is predicted by ordinary differential equations based upon the imposed spanwise pressure variation. One-step chemical kinetics are used to describe diffusion flames and multi-flame structures. The imposed normal strain rate has a significant effect on the width of downstream mixing layers as well as the burning rate. Asymptotically in the downstream direction, a constant width of the shear layer is obtained if imposed normal strain rate is constant. The one-dimensional asymptotic result is an exact solution to the multicomponent Navier-Stokes equation for both reacting and non-reacting flows, although it was obtained using the boundary-layer approximation. A similar solution with layer width growing with the square root.

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