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Diagnosing and Parameterizing the Effects of Oceanic Eddies

Part of the Mathematics of Planet Earth book series (MPE,volume 1)

Abstract

Oceanic eddies, fluctuations on scales on the order of one km to hundreds of km, derive their energy primarily from baroclinic instability processes. Currently, climate models do not incorporate the space and time variability of the effects of eddies and sub-mesoscale processes in an energy-consistent way. Eddy diffusivities are specified without connection to the energy budget and, more fundamentally, it is unclear to what extent, where and on what scales the downgradient eddy diffusion model is appropriate at all. Rotational components of the eddy fluxes associated with the advective terms in the eddy variance equation are generally large, so that production and dissipation of eddy energy do not balance locally. We will review here the current understanding of the spatial and temporal variability of eddy diffusivities and eddy–mean flow interactions that have been inferred in both observations and eddying ocean models. A focus will be on Lagrangian particle statistics as an ideal tool to describe the effects of eddies on a time mean transport and to assess the limits and validities of the eddy diffusion model. Eddy diffusivity diagnostics and the current state of eddy parameterizations in ocean models will be discussed as well as prospects for energy-consistent parameterizations.

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Fig. 6.1
Fig. 6.2
Fig. 6.3
Fig. 6.4

Notes

  1. 1.

    We refer here to transient eddies, i.e., \(T'=T-\overline{T}\) and \(\mathbf {u}'=\mathbf {u}-\mathbf {\overline{u}}\), denote the fluctuations in time due to eddies. For statistically stationary flows, time averaging is equivalent to ensemble averaging.

  2. 2.

    The divergence of the skew part of the eddy flux \(\mathbf {\nabla }\cdot \mathbf {F_{skew}}=\mathbf {\nabla }\cdot \left( \mathbf {B}\times \mathbf {\nabla } \overline{T}\right) = \left( \mathbf {\nabla } \times \mathbf {B}\right) \cdot \mathbf {\nabla } \overline{T} \equiv \mathbf {u}^{\star }\cdot \mathbf {\nabla } \overline{T}\) and hence can be expressed as advection by an eddy-driven velocity \(\mathbf {u}^{\star }=\mathbf {\nabla }\times \mathbf {B}\).

  3. 3.

    In the oceanic interior, isoneutral slopes are small, whereas in the well-mixed boundary layer of the ocean, this would not be a good approximation.

  4. 4.

    By exploiting the gauge freedom of rotational eddy flux addition, the diapycnal diffusivity \(\kappa _d\) can be defined such that \(\kappa _d=0\) follows from zero diabatic forcing \(\overline{\mathbf {Q}_b}\) as in Eden et al. (2007a).

  5. 5.

    The component of \(\mathbf {B}\) in equation 6.3 parallel to \(\mathbf {\nabla }\overline{T}\) plays no role and without loss of generality we can use the gauge condition \(\mathbf {B}_b\cdot \mathbf {\nabla }\overline{b}=0\). The solution for \(\mathbf {B}\) can be found taking \(\mathbf {F}_b \times \mathbf {\nabla }\overline{b}= -\mathbf {\nabla }\overline{b}\times \left( \mathbf {B}_b \times \mathbf {\nabla }\overline{b} \right) = -\mathbf {B}_b \left( \mathbf {\nabla }\overline{b}\cdot \mathbf {\nabla }\overline{b}\right) +\mathbf {\nabla }\overline{b}\left( \mathbf {B}_b\cdot \mathbf {\nabla }\overline{b}\right) \) and the eddy streamfunction becomes \(\mathbf {B}_b=-\left( \mathbf {F}_b \times \mathbf {\nabla } \overline{b}\right) |\mathbf {\nabla }\overline{b}|^{-2}\) (where we have neglected \(\kappa _d\) and a possible rotational flux in the eddy flux decomposition).

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Griesel, A., Dräger-Dietel, J., Jochumsen, K. (2019). Diagnosing and Parameterizing the Effects of Oceanic Eddies. In: Eden, C., Iske, A. (eds) Energy Transfers in Atmosphere and Ocean. Mathematics of Planet Earth, vol 1. Springer, Cham. https://doi.org/10.1007/978-3-030-05704-6_6

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