Wave boundary layers in rotating stratified fluid and near-inertial oscillations

Abstract

Theory of wave boundary layers (WBLs) developed by Reznik (J Mar Res 71: 253–288, 2013, J Fluid Mech 747: 605–634, 2014, J Fluid Mech 833: 512–537, 2017) is extended to a rotating stratified fluid. In this case, the WBLs arise in the field of near-inertial oscillations (NIOs) driven by a tangential wind stress of finite duration. Near-surface Ekman layer is specified in the most general form; tangential stresses are zero at the lower boundary of Ekman layer and viscosity is neglected below the boundary. After the wind ceases, the Ekman pumping at the boundary becomes a linear superposition of inertial oscillations with coefficients dependent on the horizontal coordinates. The solution under the Ekman layer is obtained in the form of expansions in the vertical wave modes. We separate from the solution a part representing NIO and demonstrate development of a WBL near the Ekman layer boundary. With increasing time t, the WBL width decays inversely proportional to $\sqrt{t}$ and gradients of fields in the WBL grow proportionally to $\sqrt{t}$; the most part of NIO is concentrated in the WBL. Structure of the WBL depends strongly on its horizontal scale L determined by scale of the wind stress. The shorter the NIO is, the thinner and sharper the WBL is; the short-wave NIO with L smaller than the baroclinic Rossby scale L_R does not penetrate deep into the ocean. On the contrary, for L ≥ L_R, the WBL has a smoother vertical structure; a significant long-wave NIO signal is able to reach the oceanic bottom. An asymptotic theory of the WBL in rotating stratified fluid is suggested.

Appendices

Appendix А

Substitution of Eq. (20a) into the r.h.s. part of Eq. (21) gives after a simple algebra:

$$ {C}_1=\frac{1}{2}\left(f-\frac{N_0^2}{f}\right)\mathbf{d}\left({A}_s-{iA}_c\right),\kern1em {C}_2=\frac{1}{4 if}\left(f-\frac{N_0^2}{f}\right)\mathbf{d}\left({A}_s+{iA}_c\right), $$

(А1a, b)

$$ {C}_3=\frac{N_0^2}{if^2}\mathbf{d}\left({A}_s\cos {ft}_0-{A}_c\sin {ft}_0+{b}_0\right). $$

(А1c)

These formulae are valid for an arbitrary stratification; N₀ is the buoyancy frequency at the boundary z =  − h_E and b₀ is the buoyancy at t = t₀.

Appendix B

It readily follows from Eq. (33d) that

$$ {\partial}_{tt}{\widehat{w}}^s={\widehat{D}}_t+\widehat{R}-{\widehat{w}}^s. $$

(В1)

Using Eq. (B1), one can rewrite the integral J_n in Eq. (32) as follows:

$$ {J}_n=\left(1-{\delta}^2\right){\int}_0^t{\widehat{w}}^s\sin {\sigma}_n\left(t-{t}^{\prime}\right)d{t}^{\prime }+{\delta}^2{\int}_0^t\left({\widehat{D}}_{t^{\prime }}+\widehat{R}\right)\sin {\sigma}_n\left(t-{t}^{\prime}\right)d{t}^{\prime }. $$

(B2)

Applying integration by parts to the integral in the first term in the r.h.s. part of Eq. (B2), one finds

$$ {F}_n={\int}_0^t{\widehat{w}}^s\sin {\sigma}_n\left(t-{t}^{\prime}\right)d{t}^{\prime }=\frac{1}{\sigma_n}{\widehat{w}}^s-\frac{1}{\sigma_n^2}{\int}_0^t{\widehat{w}}_{t^{\prime }{t}^{\prime}}^s\sin {\sigma}_n\left(t-{t}^{\prime}\right)d{t}^{\prime }, $$

whence, taking into account Eq. (B1), one obtains

$$ {F}_n=\frac{\sigma_n}{\sigma_n^2-1}\left[{\widehat{w}}^s-\frac{1}{\sigma_n}{\int}_0^t\left({\widehat{D}}_{t^{\prime }}+\widehat{R}\right)\sin {\sigma}_n\left(t-{t}^{\prime}\right)d{t}^{\prime}\right]. $$

(B3)

It follows from Eqs. (B2) and (B3) that

$$ {J}_n=\frac{\left(1-{\delta}^2\right){\sigma}_n}{\sigma_n^2-1}{\widehat{w}}_s+\frac{\delta^2{\sigma}_n^2-1}{\sigma_n^2-1}{\int}_0^t\left({\widehat{D}}_{t^{\prime }}+\widehat{R}\right)\sin {\sigma}_n\left(t-{t}^{\prime}\right)d{t}^{\prime }. $$

(B4)

Using Eqs. (B4), (30), and (32), one arrives at Eq. (34).

Appendix C

One finds from Eqs. (46) and (34) that (cf. Eqs. (40) and (41))

$$ {B}_s={K}_cF-{K}_sG+{g}_s^1,{B}_c=-\left({K}_sF+{K}_cG\right)+{g}_c^1, $$

(С1a, b)

$$ {g}_s^1=\sum \limits_{n=1}^{\infty}\frac{2}{n\pi}\sin n\pi z\left[{q}_n^1\sin \left({\sigma}_n-1\right)t+{p}_n^1\cos \left({\sigma}_n-1\right)t\right], $$

(С2a)

$$ {g}_c^1=\sum \limits_{n=1}^{\infty}\frac{2}{n\pi}\sin n\pi z\left[{p}_n^1\sin \left({\sigma}_n-1\right)t-{q}_n^1\cos \left({\sigma}_n-1\right)t\right], $$

(С2b)

$$ {p}_n^1={K}_c-{\gamma}_n{K}_c^n/{\sigma}_n,\kern1.5em {q}_n^1={K}_s-{\gamma}_n{K}_s^n/{\sigma}_n. $$

(С2c, d)

The coefficients ${p}_n^1,{q}_n^1$ are very close to the coefficient p_n, q_n in Eq. (41e, f); therefore, the functions ${g}_s^1,{g}_c^1$ are very close to g_c, − g_s in Eq. (41c, d) and can be neglected in Eq. (С1a, b).