Regularization by the Coriolis effect

Physically, ocean does not perform generalized Kuznetsov-Zakharov transformation. However, in the ocean there are finite boundaries in frequency domain. In particular, the inertial frequency, $f$, provides a truncation for the IR part of the spectrum, while the UV truncation is provided by the buoyancy frequency, $N$. These two frequencies vary from place to place, giving grounds for spectral variability. Consequently, the integrals are not truly divergent, rather they have a large numerical value dictated by the location of the IR cutoff.

All the experimental points are located in regions of the $(a,b)$ domain $a > (-b + 7)/2$ for which the integral diverges in the IR region. For most of the experimental points, the UV divergence is not an issue, as $a > -b/2 + 4$. The UV region is therefore assumed to be either sub-dominant or convergent in this section, where we study the regularization resulting from a finite value of $f$.

Since the IR cutoff is given by $f$, a frequency, it is easier to analyze the resulting integral in $(\omega _1,m_1)$ rather than in the traditional $(k_1,m_1)$ domain. Thus we need to express both the kinetic equation and the kinematic box in terms of frequency and vertical wavenumber. For this, we use the dispersion relation (12) to express $k$ in terms of $\omega$ in the description of the kinematic box (24):

where we have introduced the four curves in the $(\omega _1,m_1)$ domain that parameterize the kinematic box:

$$E_1(\omega_1) = m \frac{ -\sqrt{-f^2+\omega^2} + \sqrt{-f^2+ (\omega-\omega_1)^2}} { \sqrt{-f^2+\omega_1^2} + \sqrt{-f^2+(\omega-\omega_1)^2}} ,$$

$$E_2(\omega_1) = m \frac{ \sqrt{-f^2+\omega^2} + \sqrt{-f^2+ (\omega-\omega_1)^2}} { \sqrt{-f^2+\omega_1^2} + \sqrt{-f^2+(\omega-\omega_1)^2}},$$

$$E_3(\omega_1) = m \frac{ -\sqrt{-f^2+\omega^2} + \sqrt{-f^2+ (\omega-\omega_1)^2}} {-\sqrt{-f^2+\omega_1^2} + \sqrt{-f^2+(\omega-\omega_1)^2}} ,$$

$$E_4(\omega_1) = m \frac{ \sqrt{-f^2+\omega^2} + \sqrt{-f^2+ (\omega-\omega_1)^2}} { - \sqrt{-f^2+\omega_1^2} + \sqrt{-f^2+(\omega-\omega_1)^2}} .$$

Figure 5: The kinematic box in the $(\omega _1,m_1)$ domain. Two disconnected regions where $\omega _1 < \omega$ depict regions with ``sum'' interactions, namely $\omega =\omega _1+\omega _2,$ $m=m_1+m_2,$ type of the resonances. The connected regions where $\omega _1 > \omega$ depict ``difference'' resonances, $\omega _2=\omega _1-\omega ,$, $m_2=m_1-m$. The parameters are chosen so that $f/\omega =0.1$.

The kinematic box in the $(\omega,m)$ domain is shown in Fig. 5. To help in the transition from the traditional kinematic box to the kinematic box in $(\omega,m)$ domain, the following limits were identified:

• ID1 is the ID limit of Eq. (25b) with indices 1 and 2 being flipped
• ID2 is the ID limit of Eq. (27a) with indices 1 and 2 being flipped
• ID3 is the ID limit of Eq. (25b)
• ID4 is the ID limit of Eqs. (29b, 30b)
• PSI1 is the PSI limit of Eq. (28a)
• PSI2 is the PSI limit of Eq. (28b)
• PSI3 is the PSI limit of Eq. (26a)
• PSI4 is the PSI limit of Eq. (26b)
• ES1 is the ES limit of Eq. (25a)
• ES2 is the ES limit of Eq. (27b) with indices 1 and 2 being flipped
• ES3 is the ES limit of Eq. (27b) with indices 1 and 2 being flipped
• ES4 is the ES limit of Eq. (29a)

An advantage of the $(\omega,m)$ presentation for the kinematic box is that it allows a transparent reduction to the resonant manifold. A disadvantage is the curvilinear boundaries of the box, requiring more sophisticated analytical treatment. To proceed, we assume a power-law spectrum, similar to Eq. (2), but in the $(\omega, m$ space:

$$n_{\omega,m} \propto \omega^{\widetilde{a}} m^{\widetilde{b}}.$$ (6)

We need to transform the wave action as a function of $k$ and $m$ to a function of $\omega$ and $m$. This is done in Appendix B. The relation between $a$, $b$ and $\widetilde{a}$, $\widetilde{b}$ reads:

Equation (18) transforms into

We have used the dispersion relation $k_i=m_i\sqrt{\omega_i^2 - f^2},$ and defined $J$ as the Jacobian of the transformation from $(k_1,k_2)$ into $(\omega_1,\omega_2)$, times the $k k_1 k_2$ factor,

In Fig. 5, there are three ID corners with significant contribution to the collision integral:

1. ID1 region. In this region, $m_1$ is slightly bigger than $m$, $\omega_1$ slightly smaller than $\omega$, and $\omega_2$ and $m_2$ are both very small. This region can be obtained from the region (37b) above by interchanging indices $1$ and $2$. In this region,
   
   
   

2. ID2 region. In this region, $\omega_1$ is slightly bigger than $\omega+f$ where $\omega_2=\omega_1-\omega$ is small, and $m_2=m_1-m$ is negative and small. This is the region (37d). Also
   
   
   

3. ID3 region. Small $\omega_1$, small negative $m_1$. This corresponds to the region (37b), where
   
   
   
   Note that this region can be obtained from the region ID1 by flipping indices $1$ and $2$. Consequently, only one of the ID1, ID2 should be taken into account, with a factor of two multiplying the respective contribution.

Making these simplification, and taking into account the areas of integration in the kinematic box, we obtain

where the small parameter $\omega_{\mathrm{s}}$ is introduced to restrict the integration to a neighborhood of the ID corners. The arbitrariness of the small parameter will not affect the end result below.

To quantify the contribution of near-inertial waves to a $(\omega,m)$ mode, we write

Subsequently, near the region ID3 of the kinematic box, we write

while near the ID2 corners of the kinematic box we write

We then expand the resulting analytical expression (40) in powers of $\epsilon$ and $f$ without making any assumptions on their relative size. These calculations, including the integration over vertical wavenumbers $m_1$, are presented in Appendix C. The resulting expression for the kinetic equation is given by

$$\frac{\partial}{\partial t} n(k(\omega,m),m) = \frac{\pi}{4k} \le... ...de{b} \right) \left( \widetilde{a} - 3 \left( 3 + \widetilde{b} \right) \right)$$

$$\times m^{5 + 2 \widetilde{b}} \omega^{-3 + \widetilde{a} - \wide... ...t) }^{4 + \widetilde{a} + \widetilde{b}} (\epsilon^2+ 2 \epsilon f + 17 f^2) ~.$$

The integral over $\epsilon$ diverges at $\epsilon=0$, if $f=0$ and if $6+\widetilde{a} +\widetilde{b} > -1$.

However, if we postpone taking $f=0$ limit, we see that the integral is zero to leading order if

$$\widetilde{a} - 3\,\left( 3 + \widetilde{b} \right) =0 \quad \mathrm{or} \quad \widetilde{a} - \widetilde{b} = 0$$ (7)

or, in terms of $a$ and $b$,

$$9-2a-3b=0 \quad \mathrm{or} \quad b=0.$$ (8)

This is the family of power-law steady-state solutions to the kinetic equations dominated by infra-red ID interactions. These steady states are identical to the ID stationary states identified by mccomas-1977-82, who derived a diffusive approximation to their collision integral in the infra-red ID limit. Note that mccomas-1981 interpreted $b=0$ as a no action flux in vertical wavenumber domain, while $9-2a-3b=0$ is a constant action flux solution. What is presented in this section is a rigorous asymptotic derivation of this result. These ID solutions helps us to interpret observational data of Fig. 1 that is currently available to us.