Topographic Enhancement of Eddy Efficiency in Baroclinic Equilibration JPO, 44 (8), 2107-2126, 2014 by Ryan Abernathey Paola Cessi as told by Navid CASPO theory seminar, 28 May 2016
section 2 what s the problem? what sets the thermocline depth h? how topography affects it? Equatorward 3 km South Pole Fig. 2
section 2 topography makes thermocline shallower flat bottom ridge Fig. 4 isosurfaces of θ colors from 0 o C to 8 o C white lines: time-mean θ
section 2 meridional heat transport sets the thermocline depth quasi-adiabatic (no diapycnal mixing) 0 H(y) = 0 c p L x Z 0 Hhv( 0 )i dz = H mean + H eddy Z 0 c p L x h i + 0 c p L x hv g i dz f 0 H = c p L x y/l y + 0 c p L x h v f g (y) 0 therefore near y = L y eddy efficiency (title of paper) h = f 0 0 g vg the goal is to understand how topography affects the efficiency
section 3 model setup MITgcm, hydrostatic Boussinesq eqs. on a β-plane wind stress = 0 sin ( y/l y ) + surface θ is relaxed to θ*=8 o C y/ly with relaxation time 30 days Equatorward quasi-adiabatic interior thermocline height South Pole =2000 km Fig. 2 3 km =2000 km bottom drag r=0.0011 m 2 s -2 deformation radius=15 km
section 3 model spinup for τ0=0.2 Ν/m 2 flat bottom ridge https://vimeo.com/55486114 equilibration ~100 yr isosurfaces of θ colors from 0 o C to 8 o C
section 3 fields decomposition MEAN Standing Wave (SE) Transient (TE) A = hai(y, z) + A (x, y, z) + A 0 (x, y, z, t) time mean of the zonal mean time mean of the deviation from the zonal mean the rest Z A dx =0 Z A 0 dt =0
section 3 averages over latitude circles Vs over streamlines H(y) = 0 c p L x Z 0 Hhv( 0 )i dz = H mean + H SE + H TE = H mean + H TE H SE + H TE H TE I m feeling the heat no matter the type of average.
section 3 topography makes the thermocline shallower meridional heat transport: H(y) = 0 c p L x Z 0 Hhv( 0 )i dz = H mean + H SE + H TE = H mean + H TE (average over lat. circles) (average over streamlines) (the mean is very close to the Ekman flux calculation) h=1200 m h=1000 m higher eddy efficiency Fig. 5 flat bottom ridge
section 3 flat Vs ridge results I Fig. 6 (Munk & Palmen)
section 3 2116 JOURNAL OF PHYSICAL OCEANO flat Vs ridge results II flat ridge Fig. 78 when there is topography as wind increases the heat flux is mainly be done by the SE rather than the TE FIG. 8. Total (b a functio
section 3 flat Vs ridge results III L O F P H Y S I C A L O C E A N O G R A P H Y VOLUME 44 flat ridge Fig. 8 (Munk & Palmen) Fig. 7 FIG. Z8. Total (black) and thermal wind (red) zonal transports as Ly Z 0 Z Ly transport = a function dy dz of hui t 0 for the flat dyhu and ridge experiments. bottom i + ga Z Ly Z 0 dy 0 H 0 f 0 0 H dz @h i @y z {z } TW
section 4 a simplified 2-layer QG model PV at each layer q 1 = r 2 1 + F 1 ( 2 1 ) q 2 = r 2 2 + F 2 ( 1 2 ) + f 0 H 2 h b decompose the flow fields into: j(x, y, t) = h j i(y) + j (x, y) + 0 (x, y, t) j =1, 2 MEAN Standing Wave (SE) Transient (TE)
section 4 a simplified 2-layer QG model we get then an equation for the TE (@ t + U 1 @ x )(q 1 + q0 1)+J( 1 + 0 1,q 1 + q0 1)+@ y hq 1 i@ x ( 1 + 0 1)= (@ t + U 2 @ x )(q 2 + q0 2)+J( 2 + 0 2,q 2 + q0 2)+@ y hq 2 i@ x ( 2 + 0 2)= which if we time average gives us the equations for the standing wave component @ y 0 H 1 r h r 2 ( 2 H + 2) 0 2 @ y U 2 i U 1 @ x q 1 + @ yhq 1 i@ x 1 + J( 1,q 1 )+J( 0 1,q0 1 ) @ y h(@ x 1 )q 1 i @ yh(@ x 0 1)q 0 1i =0 where did the wind stress go? U 2 @ x q 2 + @ yhq 2 i@ x 2 + J( 2,q 2 )+J( 2 0,q0 2 ) @ y h(@ x 2 )q 2 i @ yh(@ 0 x 2)q2i 0 r = r 2 2 H 2
section 4 a simplified 2-layer QG model @ x U, @ y U =) neglect @ y neglect terms quadratic in parametrize J( 0 j,q0 j )= K r2 q j U 1 @ 2 x 1 + U 1F 1 2 + 1 F 1 U 2 1 = K@ xq 1 U 2 @ 2 x 2 + U 2F 2 1 + 2 F 2 U 1 2 = K@ xq 2 r H 2 @ x 2 f 0 H 2 U 2 h b Eqs (30)-(31) h s H 1 U 1 1 H 2 U 2 1 2 σ h b
section 4 the standing wave components U 1 @ 2 x 1 + U 1F 1 2 + 1 F 1 U 2 1 = K@ xq 1 U 2 @ 2 x 2 + U 2F 2 1 + 2 F 2 U 1 2 = K@ xq 2 r H 2 @ x 2 f 0 H 2 U 2 h b (ridge>deformation) 1 p F2 (K is small or TE effect negligible) (Rhines scale = ridge) K U 2 1 s U 1 U 2 1 U 1 2 using this approximation one can proceed to calculate
section 4 the heat transport in the QG model Z H h i H mean + H SE + H TE H h 1 @ x 2 i + h 0 1@ x 0 2i = Kf(U 1 U 2 ) 1+ h(@ x 2 )2 i U 2 2 H eddy heat transport is augmented by the presence of the standing wave ψ2 due to the topography!
section 4 and the thermocline slope in the QG model s = f 0 U 1 U 2 g 0 = 0/( 0 Kf 0 ) 1+ h(@ x 2 )2 i U 2 2 the planetary scale slope isopycnal is reduced due to the standing wave ψ2
section 4 the QG model captures the qualitative behavior AUGUST 2014 A B E R N A T H E Y A N D C E S S I Fig. 9 FIG. 10. The to f 0 (c y 1 2 cy 2 ) solution given by K 5 2300 m
section 5 local cross-stream heat fluxes H 0 = 0 cp 2120 Z < 0 r F dx dy, F= Z 0 u dz = H Z 0 u + u0 0 dz H JOURNAL OF PHYSICAL OCEANOGRAPHY = Fmean + Feddy VOLUME 44 div Fdiv mean + Feddy flat bottom the ridge enhances diffusivity near it but suppresses it overall resulting in smaller mean diffusivity ridge Fig. 11 FIG. 11. (left) The local eddy heat transport Fdiv (x, y). The arrows show the direction and magnitude of the
section 6 how fast the eddies are carried by the flow? flat bottom ridge eddies propagate much faster without topography Fig. 13 authors argue that eddy production is done through convective instability for the flat case and through absolute instability for the ridge case
section 6 convective Vs absolute instability see https://vimeo.com/55486114
section 6 local cross-stream heat fluxes flat bottom ridge conversion from PE to KE Fig. 14 EKE is produced mainly downstream the ridge
discussion We never clearly see the SE in the full model (only its signature in the Hovmöller diagram) H mean H Ekman Why? H SE H TE averaging-over-streamlines picture? as wind stress increases. How this translates to the The inability of existing parameterizations to account for local instability and nonlocal eddy life cycles constitutes the main obstacle toward a more complete theory of baroclinic equilibration in the presence of large topography and the more general problem of inhomogenous geostrophic turbulence.