Conditions for Transformation of a Mesoscale Vortex into a Submesoscale Vortex Filament When the Vortex Is Stretched by an Inhomogeneous Barotropic Flow

Abstract

In this paper, we study the effects of strong stretching in the horizontal plane of large-scale mesoscale ocean eddies using the theory of ellipsoidal eddies in the World Ocean. The objective is to theoretically determine the physical conditions for unbounded stretching eddies and also check the feasibility of these conditions in the ocean. We estimate the share of mesoscale ocean eddies that are elongated into filaments and then they redistribute energy from the mesoscale to the submesoscale.

APPENDIX

The starting point of the study is a system of ordinary differential equations describing the behavior of an ellipsoidal vortex (see formulas (12a) and (12b)):

$$\left\{ \begin{gathered} \dot {\varepsilon } = 2\varepsilon e{\text{cos}}2\theta {\text{,}} \\ \dot {\theta } = \frac{{\sigma K}}{2}\int\limits_0^\infty {\frac{{\mu \partial \mu }}{{\sqrt {{{{\left( {{{\mu }^{2}} + \left( {\varepsilon + \frac{1}{\varepsilon }} \right)\mu + {\text{1}}} \right)}}^{3}}\left( {{{K}^{2}} + \mu } \right)} }}} \\ + \,\,\gamma - \frac{{{{\varepsilon }^{2}} + 1}}{{{{\varepsilon }^{2}} - 1}}e{\text{sin}}2\theta \\ \end{gathered} \right\}.$$

(16)

The solution to this system is a family of integral curves:

$$\begin{gathered} {\text{sin}}2\theta \left( \varepsilon \right) = \frac{\sigma }{e}\left( {S\frac{\varepsilon }{{{{\varepsilon }^{2}} - 1}}} \right) \\ + \,\,\frac{\gamma }{\sigma }\frac{{\varepsilon - 1}}{{\varepsilon + 1}} + \frac{\varepsilon }{{{{\varepsilon }^{2}} - 1}}\int\limits_1^\varepsilon {\frac{{{{\varphi }^{2}} - 1}}{{{{\varphi }^{2}}}}} \\ \times \,\,\frac{K}{2}\int\limits_0^\infty {\frac{{\mu \partial \mu }}{{\sqrt {{{{\left( {{{\mu }^{2}} + \left( {\varphi + \frac{1}{\varphi }} \right)\mu + {\text{1}}} \right)}}^{3}}\left( {{{K}^{2}} + \mu } \right)} }}d\varphi {\text{,}}} \\ \end{gathered} $$

(17)

where S is the integration constant. Detailed analysis of this family of integral curves showed that for K > 0 and $\frac{\gamma }{\sigma } \in R$, there are six different types of patterns of integral curves, which, with the selection of parameter $\frac{\gamma }{\sigma }$, can be obtained for any K.

In the course of further research, the value of the parameter K was fixed and the corresponding types of integral curves were determined, analysis of which made it possible to construct in the parameter plane $\left( {\frac{\gamma }{\sigma },\frac{e}{\sigma }} \right)$ zones corresponding to different modes of vortex behavior. Then, a similar procedure was carried out for another value of parameter K, when the zones of behavior of vortices were constructed for all K values of interest; transformation was made from the parameter plane $\left( {\frac{\gamma }{\sigma },\frac{e}{\sigma }} \right)$ to the parameter plane $\left( {\frac{\sigma }{e},\frac{\gamma }{e}} \right)$. The result of such a transformation is a map of the zones of behavior of ellipsoidal vortices shown in Figs. 1 and 2.