Turbulent Energy Cascade in the Gulf of Mexico
Authors: Ma Yinxiang , Hu Jianyu , Huang Yongxiang 
Published: March 2022

&lt;p&gt;Due &amp;#160;to the extreme complexity of the oceanic dynamics, e.g., stratification, air-sea interaction, &amp;#160;waves, current, tide, etc., the corresponding turbulent cascade remains unknown. The third-order longitudinal structure-function is often employed to diagnose &amp;#160;the cascade direction and intensity, which is written as&amp;#160; S&lt;sub&gt;LLL&lt;/sub&gt;(r)=&lt; &amp;#916; u&lt;sub&gt;L&lt;/sub&gt;&lt;sup&gt;3&lt;/sup&gt;(r)&gt;, where &amp;#916; u&lt;sub&gt;L&lt;/sub&gt; is the&amp;#160; velocity increment along the distance vector &lt;strong&gt;r&lt;/strong&gt;, and r is the modulus of &lt;strong&gt;r&lt;/strong&gt;. In the case of&amp;#160; three-dimension homogeneous and isotropic turbulence, S&lt;sub&gt;LLL&lt;/sub&gt;(r) is scaled as -4/5&amp;#949;r in the inertial range, where &amp;#949; is the energy dissipation rate per unit. &amp;#160;In this work, S&lt;sub&gt;LLL&lt;/sub&gt;(r) is estimated for two experimental velocities that obtained in the Gulf of Mexico, namely Grand LAgrangian Deployment (GLAD) and the LAgrangian Submesoscale ExpeRiment (LASER). The experimental S&lt;sub&gt;LLL&lt;/sub&gt;(r) for both experiments shows a transition from negative values to a positive one roughly at r&lt;sub&gt;T&lt;/sub&gt;=10km, corresponding to a timescale &amp;#160;around &amp;#964;&lt;sub&gt;T&lt;/sub&gt;=12-hour (e.g., &amp;#964;&lt;sub&gt;T&lt;/sub&gt;=r&lt;sub&gt;T&lt;/sub&gt;/u&lt;sub&gt;rms&lt;/sub&gt; with u&lt;sub&gt;rms &lt;/sub&gt;&amp;#8776;0.24m/s. &amp;#160;Power-law is evident for the scale on the range 0.01&amp;#8804; r&amp;#8804;1km as S&lt;sub&gt;LLL&lt;/sub&gt;(r)&amp;#8764; -r&lt;sup&gt;1.45&amp;#177;0.10&lt;/sup&gt;, and for the scale on the range 30&amp;#8804; r&amp;#8804;300km as S&lt;sub&gt;LLL&lt;/sub&gt;(r)&amp;#8764; r&lt;sup&gt;1.45&amp;#177;0.10&lt;/sup&gt;. Note that a weak stratification with depth of 10&amp;#8764;15m has been reported for the GLAD experiment, indicating a quasi-2D flow topography. The scaling ranges are above this stratification depth. Hence, the famous Kraichnan's 2D turbulence theory or the geostrophic turbulence proposed by Charney are expected to be applicable. However, due to the complexity of real oceanic flows, hypotheses behind these theories cannot be verified either directly or indirectly. To simplify the situation, we still consider here the sign of &amp;#160;S&lt;sub&gt;LLL&lt;/sub&gt;(r) as an indicator of the energy cascade. It thus suggests a possible forward energy cascade below the spatial scale r&lt;sub&gt;T&lt;/sub&gt;, and an inverse one above the scale &amp;#160;spatial r&lt;sub&gt;T&lt;/sub&gt;. &amp;#160;While, the scaling exponents 1.45 are deserved more studied in the future if more data is available.&lt;/p&gt;&lt;p&gt;&amp;#160;&lt;/p&gt;&lt;p&gt;&lt;strong&gt;Ref.&lt;/strong&gt;&lt;/p&gt;&lt;p&gt;Charney, J. G. (1971). Geostrophic turbulence. J. Atmos. Sci., 28(6), 1087-1095.&lt;/p&gt;&lt;p&gt;Frisch, U., &amp; Kolmogorov, A. N. (1995). Turbulence: the legacy of AN Kolmogorov. Cambridge University Press.&lt;/p&gt;&lt;p&gt;Alexakis, A., &amp; Biferale, L. (2018). Cascades and transitions in turbulent flows. Phys. Rep., 767, 1-101.&lt;/p&gt;&lt;p&gt;Dong, S., Huang, Y., Yuan, X., &amp; Lozano-Dur&amp;#225;n, A. (2020). The coherent structure of the kinetic energy transfer in shear turbulence. J. Fluid Mech., 892, A22.&lt;/p&gt;&lt;p&gt;Poje, A. C., &amp;#214;zg&amp;#246;kmen, T. M., Bogucki, D. J., &amp; Kirwan, A. D. (2017). Evidence of a forward energy cascade and Kolmogorov self-similarity in submesoscale ocean surface drifter observations. Phys. Fluids, 29(2), 020701.&lt;/p&gt;