The seasonal cycle of submesoscale flows

The seasonal cycle of submesoscale flows in the upper ocean is investigated in an idealised model domain analogous to mid-latitude open ocean regions. Submesoscale processes become much stronger as the resolution is increased, though with limited evidence for convergence of the solutions. Frontogenetical processes increase horizontal buoyancy gradients when the mixed layer is shallow in summer, while overturning instabilities weaken the horizontal buoyancy gradients as the mixed layer deepens in winter. The horizontal wavenumber spectral slopes of surface temperature and velocity are steep in summer and then shallow in winter. This is consistent with stronger mixed layer instabilities developing as the mixed layer deepens and energising the submesoscale. The degree of geostrophic balance falls as the resolution is made finer, with evidence for stronger non-linear and high-frequency processes becoming more important as the mixed layer deepens. Ekman buoyancy fluxes can be much stronger than surface cooling and are locally dominant in setting the stratification and the potential vorticity at fronts, particularly in the early winter. Up to 30% of the mixed layer volume in winter has negative potential vorticity and symmetric instability is predicted inside mesoscale eddies as well as in the frontal regions outside of the vortices.

Enstrophy is also dissipated in the momentum equation using adaptive viscous 85 schemes first developed by Smagorinsky (1963), Leith (1996)  and reduced by a factor of four for each doubling in resolution. A partial-slip 96 bottom boundary condition is imposed with a quadratic bottom drag (Arbic 97 and Scott, 2008) using a non-dimensional quadratic drag coefficient of 3×10 −3 .

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In addition, vertical mixing of both heat and momentum is carried out with a 99 Laplacian operator with a constant diffusion coefficient of 4×10 −5 m 2 s −1 . The 100 mixed layer depth is defined throughout as the first depth where the temperature 101 difference from the surface is greater than 0.1 • C.

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The advection of temperature is carried out using the Prather scheme (Prather,103 1986). This is an upwind scheme that conserves second-order moments in sub-  to capture the effect of vertical convective mixing (Marshall and Schott, 1999).

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The KPP scheme also applies higher diffusive coefficients in the event of negative 124 stratification, even if this is not associated with destabilising surface buoyancy 125 forcing as can occur in the presence of down-front winds. In these cases of static 126 instability the KPP scheme applies a high (5×10 −3 m 2 s −1 ) vertical diffusion 127 coefficient rather than instantaneously mixing buoyancy as done by the default     sin (kx + φ 1 (k, l)) sin (ly + φ 2 (k, l)) ,  The averaging operator denoted by an overbar is a horizontal average over a model level: g(x, t) = 1 A x y gdxdy, where g is an arbitrary function, x is the position vector, t is time and A is the 197 horizontal area.

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The overall buoyancy and momentum fields are compared at different reso-          wind balance that includes the full non-linear advective terms: where u = (u, v, w) is the velocity vector and µ adv is adapted from µ geo to 324 include the contribution of the advective terms. A similar notation is adopted 325 for this term in the balances below. The advection terms include the centripetal 326 acceleration and so this non-linear balance may better describe the force balance 327 in vortices and at curved fronts.

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The model solution also supports internal waves that lead to more rapid where K is the vertical viscous coefficient that is set by the KPP scheme in 341 the mixing layer but is a constant below and τ z is the wind stress divergence (3.5) The annual cycle in geo is shown in Fig. 11. This shows that the degree of 348 balance falls as the resolution is made finer, both in the mixed layer and in the 349 interior. Vertically, the degree of balance is lower in the mixed layer than in 350 the interior, though minima are often found at the base of the deepening mixed 351 layer.

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While geostrophic balance is the primary balance, there is a change in the 353 residual mean balance across this range of resolutions. Figure 12 shows the ver-   where:  Ekman buoyancy flux (EBF) and can be diagnosed as: where τ is the wind stress, ρ o is a reference density and k is the unit vertical

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The effect of the EBF is investigated further in Section 4.3.
where ζ g = f + ∇ × u g and u g is the geostrophic velocity. When -180 • < φ Ri b < 457 −135 • , the potential vorticity is negative due to unstable stratification and 458 convective instability is expected to dominate. When -135 • < φ Ri b < −90 • , the 459 potential vorticity is negative due to both unstable stratification and horizontal 460 buoyancy gradients and so a hybrid convective/symmetric mode is predicted.

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For stable stratification and cyclonic vorticity -90 implies that a symmetric instability should arise. For anti-cyclonic vorticity  winter, less symmetric instability is expected.

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A similar analysis can be carried out in Fig. 17 where the potential vorticity   The second panel in Figure 19 shows that vertical buoyancy fluxes averaged over 532 the mixed layer become stronger as the resolution becomes finer and has its peak where H is the mixed layer depth. This is shown in Figure 19 (top panel) 558 where the seasonal cycle in APE is somewhat different that that of the vertical

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To follow on from this work, the presence of submesoscale filaments inside 652 mesoscale vortices will be examined in more detail (Brannigan, in prep.). The 653 development of stratification in the model as the resolution varies will also be 654 investigated to illustrate why a deeper mixed layer develops at finer resolution.

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These predictions will also be tested with the OSMOSIS mooring array from 656 the North Atlantic.