Excitation mechanism of near-inertial waves in baroclinic tidal flow caused by parametric subharmonic instability

Abstract

Parametric subharmonic instability (PSI) transfers energy from low-mode semidiurnal baroclinic tidal flow to high-mode near-inertial waves at latitudes ∼30°, inducing strong ocean mixing and hence affecting the global ocean circulation. Nevertheless, intuitive descriptions of the physical mechanism for energy transfer by PSI are very sparse. In this study, we reformulate this phenomenon to present a visual image of its mechanism based on a combination of simple classical theories such as beats and parametric excitation without adhering to a strict mathematical formula. It is shown that two small-scale near-inertial waves with slightly different wavenumbers propagating in opposite directions superpose to create beats. When the resulting beats have the peak-to-peak length and the phase velocity equal to the wavelength and the phase velocity of large-scale semidiurnal baroclinic tidal flow, respectively, continuous acceleration of near-inertial motions takes place under the effects of convergence and horizontal shear of the background semidiurnal baroclinic tidal flow. The resonant condition for PSI can thus be easily understood by introducing the well-known concept of beats which also provides a natural explanation for the large difference in spatial scales between the semidiurnal baroclinic tidal flow and the resulting near-inertial waves.

Appendix

1.1 A. Order estimation of the energy production term

The energy production G in (7) can be reduced to (10) by algebraic manipulation as follows.

For inertial gravity waves, (8) and (9) yield

$$ \frac{k}{m}\sim \left|\frac{w}{u}\right|\sim \left|\frac{\omega b}{N^2u}\right|\sim \sqrt{\frac{\omega^2-{f}^2}{N^2-{\omega}^2}}. $$

(A1)

In our study, the frequencies of the background and fluctuating components are about 2f and f, respectively, so that, for f < < N, (A1) gives the relation between wavenumbers for the background wave k ₀, m ₀ and fluctuating waves k ′, m ′ such that

$$ \frac{k^{\prime }}{m^{\prime }}<<\frac{k_0}{m_0}\sim \frac{\sqrt{3}f}{N}<<1. $$

(A2)

Combining (A1) and (A2), we can obtain

$$ \begin{array}{l}\left|G\right|=\left|\left(u^{\prime }{k}_0+w^{\prime }{m}_0\right)\left(u^{\prime }U+v^{\prime }V+w^{\prime }W+\frac{b^{\prime }B}{N^2}\right)\right|\hfill \\ {}\sim \left|u^{\prime }{k}_0\left(u^{\prime }U+v^{\prime }V\right)\right|\hfill \\ {}=\left|u{\prime}^2\frac{\partial U}{\partial x}+u^{\prime }v^{\prime}\frac{\partial V}{\partial x}\right|\hfill \end{array}. $$

(A3)

1.2 B. Derivation of the growth rate of near-inertial beats

We give the analytical formulation of PSI using the theory of parametric excitation explained in Sect. 2.1.

Since the terms of advection by background flow have nothing to do with energy production, (5) can be rewritten as

$$ \left\{\begin{array}{l}\frac{\partial \mathbf{u}^{\prime }}{\partial t}+\mathbf{u}^{\prime}\cdot \nabla \mathbf{U}+f{\mathbf{e}}_z\times \mathbf{u}^{\prime }=-\nabla p^{\prime }+b^{\prime }{\mathbf{e}}_z\\ {}\nabla \cdot \mathbf{u}^{\prime }=0\\ {}\frac{\partial b^{\prime }}{\partial t}+\mathbf{u}^{\prime}\cdot \nabla B+w^{\prime }{N}^2=0.\end{array}\right. $$

(B1)

Substituting the background field expression and beats solutions

$$ \left\{\begin{array}{l}\left(U,V,W,B\right)=\left({U}_0^{+},{V}_0^{+},{W}_0^{+},{B}_0^{+}\right) \exp i\left({k}_0x+{m}_0z-{\omega}_0t\right)+c.c.\\ {}{\left(u\prime, v\prime, w\prime, p\prime, b\prime \right)}^T={\mathbf{q}}_{\mathbf{1}} \exp i\left({k}_1x+{m}_1z-{\omega}_1t\right)+{\mathbf{q}}_{\mathbf{2}} \exp i\left({k}_2x+{m}_2z+{\omega}_2t\right)\end{array}\right. $$

(B2)

with resonant condition ω ₁ + ω ₂ = ω ₀ and k ₁ − k ₂ = k ₀ into (B1) and neglecting all the terms except those with wavenumbers k ₁, k ₂, we can get the algebraic equations

$$ \left\{\begin{array}{c}\hfill {\mathbf{A}}_{1+}{\mathbf{q}}_1={\mathbf{B}}_{2+}{\mathbf{q}}_2\hfill \\ {}\hfill {\mathbf{A}}_{2-}{\mathbf{q}}_2={\mathbf{B}}_{1-}{\mathbf{q}}_1\hfill \end{array}\right., $$

(B3)

where

$$ \begin{array}{c}\hfill {\mathbf{A}}_{j\pm}\equiv \left[\begin{array}{ccccc}\hfill \mp i{\omega}_j\hfill & \hfill -f\hfill & \hfill 0\hfill & \hfill i{k}_j\hfill & \hfill 0\hfill \\ {}\hfill f\hfill & \hfill \mp i{\omega}_j\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill \mp i{\omega}_j\hfill & \hfill i{m}_j\hfill & \hfill -1\hfill \\ {}\hfill i{k}_j\hfill & \hfill 0\hfill & \hfill i{m}_j\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill {N}^2\hfill & \hfill 0\hfill & \hfill \mp i{\omega}_j\hfill \end{array}\right]\kern1em \mathrm{and}\kern14.62em \hfill \\ {}\hfill {\mathbf{B}}_{j\pm}\equiv \left[\begin{array}{ccccc}\hfill \mp i{k}_0{U}_0^{\pm}\hfill & \hfill 0\hfill & \hfill \mp i{m}_0{U}_0^{\pm}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill \mp i{k}_0{V}_0^{\pm}\hfill & \hfill 0\hfill & \hfill \mp i{m}_0{V}_0^{\pm}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill \mp i{k}_0{W}_0^{\pm}\hfill & \hfill 0\hfill & \hfill \mp i{m}_0{W}_0^{\pm}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill \mp i{k}_0{B}_0^{\pm}\hfill & \hfill 0\hfill & \hfill \mp i{m}_0{B}_0^{\pm}\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right]\kern1em \left(j=1,2;\ \mathrm{double}\ \mathrm{sign}\ \mathrm{corresponds}\right).\kern2em \hfill \end{array} $$

(B4)

In these equations, superscripts “±” represent complex conjugate. In order for (B3) to have nontrivial solutions, ω ₁ and ω ₂ need to satisfy the following conditions:

$$ \left|\begin{array}{cc}\hfill {\mathbf{A}}_{1+}\hfill & \hfill -{\mathbf{B}}_{2+}\hfill \\ {}\hfill -{\mathbf{B}}_{1-}\hfill & \hfill {\mathbf{A}}_{2-}\hfill \end{array}\right|=0\kern0.5em \mathrm{and}\kern0.5em {\omega}_1+{\omega}_2={\omega}_0. $$

(B5)

This is one of the generalized eigenvalue problems, easily calculated with numerical methods.