Bouchut F, Le Sommer J, Zeitlin V
Département de Mathématiques et Applications, Ecole Normale Supérieure, 45 rue d'Ulm, 75005 Paris, France.
Chaos. 2005 Mar;15(1):13503. doi: 10.1063/1.1857171.
A clear-cut signature of a wave-breaking event is irreversible modification of the mean flow. In this paper, we provide examples of different breaking mechanisms and show that breaking scenario of equatorial waves in the beta-plane shallow water model is determined by the degree of balance between the zonal component of the Coriolis force and the pressure gradient. Our analysis is based on a specially designed numerical method which guarantees two essential conditions to simulate nonlinear equatorial waves: (i) the scheme converges toward weak solutions including shocks and (ii) preserves the steadiness of balanced stationary solutions. This allows for accurate diagnostics of Lagrangian invariants of motion such as passive tracer density or potential vorticity. For unbalanced waves, the lack of balance leads to shock formation in finite time. In shock fronts, the variation of the dissipation rate induces a nonadvective potential vorticity flux and violates the local potential vorticity conservation valid for smooth solutions. This dissipative breaking mechanism is generic for unbalanced waves and is associated with enhanced mixing. For long, balanced (Rossby) waves, breaking consists in appearance of recirculation regions. It results in the formation of propagating patterns, the equatorial modons, which trap fluid particles. Such breaking occurs during the propagation of Rossby wave packets with positive geopotential anomaly and is strengthened by decreasing fluid depth. The modons are robust and collide quasielastically with Kelvin waves.
波破碎事件的一个明确特征是平均流的不可逆改变。在本文中，我们给出了不同破碎机制的示例，并表明β平面浅水模型中赤道波的破碎情形由科里奥利力的纬向分量与压力梯度之间的平衡程度决定。我们的分析基于一种专门设计的数值方法，该方法保证了模拟非线性赤道波的两个基本条件：（i）该格式收敛于包括激波在内的弱解；（ii）保持平衡定常解的稳定性。这使得能够精确诊断拉格朗日运动不变量，如被动示踪剂密度或位涡。对于非平衡波，缺乏平衡会导致在有限时间内形成激波。在激波前沿，耗散率的变化会引起非平流位涡通量，并违反光滑解所适用的局部位涡守恒。这种耗散破碎机制对于非平衡波是普遍存在的，并且与增强的混合相关。对于长的、平衡的（罗斯贝）波，破碎表现为再循环区域的出现。这导致了传播模式——赤道模的形成，赤道模会捕获流体粒子。这种破碎发生在具有正位势异常的罗斯贝波包传播过程中，并随着流体深度的减小而增强。赤道模很稳健，并且与开尔文波发生准弹性碰撞。
