Momentum Budget

Integral momentum budget between surface and critical layer

To interpret the growth rate modification, we formulate a momentum budget in the full Lagrangian coordinates. It emphasizes that the critical-layer interaction is controlled by the coupling between the total phase speed $c$ and the total Lagrangian mean flow $U$, not just the background/mean Eulerian velocity.

$ \frac{\partial}{\partial t} \displaystyle\int_{0}^{b_c} \mathcal{J} \,\dot{x}\,\mathrm{d}b $ $ + $ $ \frac{\partial}{\partial a} \displaystyle\int_{0}^{b_c} p\,z_b\,\mathrm{d}b $ $ = $ $ \rho_0 g z\,\dot{z} \Big|_{b=0}^{b=b_c} $ $ - $ $ \frac{1}{2}\rho_0 \big(\dot{x}^2+\dot{z}^2\big)\dot{z} \Big|_{b=0}^{b=b_c} $ $ + $ $ \rho_0 f(b,t)\Big|_{b=0}^{b=b_c} $

Hover or tap a term in the equation to see its role in the momentum budget.

How the budget explains growth-rate suppression

The pressure representation makes the dependence on the total Lagrangian mean velocity (the Lagrangian drift) explicit:

\[ p = \rho_0\left[ -g z + \frac{1}{2}\,\underbrace{(c-U)}_{\text{total relative speed}} \big(\dot{x}^2+\dot{z}^2\big) + f(b,t) \right]. \]

The momentum budget therefore links the form drag directly to the coupling between the total phase speed $c$ and the total Lagrangian mean flow $U$ throughout the layer between the interface and the critical level, where we now have the understanding that the critical level is only necessarily "critical" with respect to the leading-order terms, $c^{(0)}=U^{(0)}$.

At leading-order, we have the condition $c^{(0)}(b_c) = U^{(0)}(b_c)$ (written here in Lagrangian coordinates), which makes the momentum transfer from air to sea maximally efficient.
As the wave-induced mean flow $U^{(2)}$ grows with steepness, the total phase speed and total Lagrangian mean flow no longer match: in general $c \neq U$.
This mismatch detunes the resonance at the critical level, reducing the efficiency of momentum transfer and the net form drag in the momentum budget.

The momentum budget and the growth-rate calculation give a consistent picture: the wave-induced mean flow modifies the coupling between phase speed and Lagrangian mean flow at the critical level, so that increasing steepness leads to a suppression of wave growth.