Turbulent Energy Cascade in the Gulf of Mexico

Due  to the extreme complexity of the oceanic dynamics, e.g., stratification, air-sea interaction,  waves, current, tide, etc., the corresponding turbulent cascade remains unknown. The third-order longitudinal structure-function is often employed to diagnose  the cascade direction and intensity, which is written as  S_LLL(r)=< Δ u_L³(r)>, where Δ u_L is the  velocity increment along the distance vector r, and r is the modulus of r. In the case of  three-dimension homogeneous and isotropic turbulence, S_LLL(r) is scaled as -4/5εr in the inertial range, where ε is the energy dissipation rate per unit.  In this work, S_LLL(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_LLL(r) for both experiments shows a transition from negative values to a positive one roughly at r_T=10km, corresponding to a timescale  around τ_T=12-hour (e.g., τ_T=r_T/u_rms with u_rms ≈0.24m/s.  Power-law is evident for the scale on the range 0.01≤ r≤1km as S_LLL(r)∼ -r^1.45±0.10, and for the scale on the range 30≤ r≤300km as S_LLL(r)∼ r^1.45±0.10. Note that a weak stratification with depth of 10∼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  S_LLL(r) as an indicator of the energy cascade. It thus suggests a possible forward energy cascade below the spatial scale r_T, and an inverse one above the scale  spatial r_T.  While, the scaling exponents 1.45 are deserved more studied in the future if more data is available.

 

Ref.

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Dong, S., Huang, Y., Yuan, X., & Lozano-Durán, A. (2020). The coherent structure of the kinetic energy transfer in shear turbulence. J. Fluid Mech., 892, A22.

Poje, A. C., Özgökmen, T. M., Bogucki, D. J., & 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.