等熵大气中的非线性扰动与弱激波
Nonlinear perturbations and weak shock waves in isentropic atmospheres
AI总结：
针对恒星包层声学扰动演化为弱激波的问题，利用等熵平面平行Euler方程的精确Riemann不变量获得解析解，并结合Whitham近似求解激波动力学，在中等马赫数范围内与数值模拟结果高度吻合。
摘要：
对恒星包层的声学扰动可通过非线性波陡峭化导致弱激波的形成。在恒星表面附近，弱激波强度增加，并可能导致部分恒星包层的抛射。尽管在小振幅波或强激波极限下存在精确的流体方程解析解，但连接这些阶段通常需要模拟。我们利用以下事实来解决这一问题：在恒定引力场存在时，当流动为等熵时，平面平行 Euler 方程允许精确的 Riemann 不变量。我们获得了声学扰动的精确解，并表明在它们陡峭化为激波后，可使用 Whitham 近似通过简单的常微分方程求解弱至中等强度区间内的激波动力学。数值模拟表明，我们的解析激波近似在中等（约几至15）马赫数范围内是准确的，且精度随绝热指数增加而提高。
英文摘要：
Acoustic perturbations to stellar envelopes can lead to the formation of weak shock waves via nonlinear wave-steepening. Close to the stellar surface, the weak shock wave increases in strength and can potentially lead to the expulsion of part of the stellar envelope. While accurate analytic solutions to the fluid equations exist in the limits of low amplitude waves or strong shocks, connecting these phases generally requires simulations. We address this problem using the fact that the plane parallel Euler equations, in the presence of a constant gravitational field, admit exact Riemann invariants when the flow is isentropic. We obtain exact solutions for acoustic perturbations and show that after they steepen into shock waves, Whitham's approximation can be used to solve for the shock's dynamics in the weak to moderately strong regimes, using a simple ordinary differential equation. Numerical simulations show that our analytic shock approximation is accurate up to moderate ($\sim$ few--15) Mach numbers, where the accuracy increases with the adiabatic index.
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