Ed saturation in a barotropic model Navid C. Constantinou & William R. Young Scripps Institution of Oceanograph, UC San Diego Can a barotropic QG model ehibit ed saturation? What is the minimum requirements Animated view of the Southern Ocean topograph b V. Tamsitt. for ed saturation? 21 st AOFD Conference Portland, June 26th, 217
the plan stea mean zonal wind stress Revisit an old barotropic QG model on a β-plane. F {{ (art 1979, Dave 198, Bretherton & aidvogel 1976, ollowa 1987, Carnevale & Fredericksen 1987) depth h(, ) ẑ ŷ ˆ A distinctive feature of this model is a large-scale barotropic zonal flow U(t). Stu how momentum is balanced b this is the ACC topographic PV total streamfunction η f h U(t) ψ(,, t) topographic form stress and investigate the QGPV q r 2 requirements for ed saturation.
al stress F. The dissipation in (??) is due to bottom drag with coefficient η)t U(ζthe June η) J(ψ, ζ sort η) of scale selective dissipation 2, 217 it(ζν.presumabl model has some ebruar ruar 16,16, 217 217 (1) (2) βψqg µζ hper visc a barotropic model for mid-ocean region topograph in the Southern Ocean total streamfunction total streamfunction U(t) ψ(,, t) topographic contribution to potential vorticit and 2 q r total U(t) ψ(,, t) QGPV: q ψ η β ψstreamfunction 2 2! "# $ ζ ( of) QGPV ψ, Material conservation QGPV odel! ζ "# q ψ2 ψ η β ) J(, r ) η) η) J(ψ, J(ψ, ζr ζ η) tη) U (r (1) (1)! "# $ f h ζ depth h(, ) η 2 βψ βψ µζ hper hper visc visc (2) (2) µζ µ r hper visc. 2 2 QGPV: $ (3) eq2 (4) l (1) (2) depth atlarge-scale San h(, ) a flow of Oceanograph, Universit of California Diego, Jolla, CA 9293 23, t. Following CF87, there is a U(t) determined b ribution ution to to potential potential vorticit vorticit and and arge-scale zonal momentum 2 2 ζ ( (! 2 2 ) ψ ),ψ, F t $ )U!"# "#$ ( U ( ) J( t topographic µu ψη verticall integrated and f h eq2 PV:(3)(3) ηhorizontall eq2 averaged, ) zonal angular momentum equation 1, µ topographic hper visc. (3) (4) a mid-ocean region eq3 na o z n a sf e m res t d s a d ste win domain average ge over the domain and σ(t) is the form stress. The large-scale flow is form stress size 2π 2π F87,, there there is ais large-scale a large-scale flow flow U(t) U(t) determined determined b b hestress topographic contribution to potential al F. The dissipation in (??)τsvorticit is due toand bottom drag with coefficient F F µu µu ψη ψη, domain, eq3eq3 F (4):(4) is average wind stress forcing periodic boundar conditions ρsome t ν. Presumabl the model has sort of scale selective dissipation 2 2 stress. uthe (@ @ ), (5) eq2 ain and and σ(t) σ(t) is the is the form form stress. The large-scale large-scale flow flow is is (art 1979, Dave 198, Bretherton & aidvogel 1976, ollowa 1987, Carnevale & Fredericksen 1987) t (uu) f v (p ) τz credit: V. Tamsitt, Sc
Question: Does this barotropic QG model show ed saturation? Do we need baroclinicit? Do we even need channel walls?
what is ed saturation? The insensitivit of the total ACC volume transport to wind stress increase. Munda, Johnson & Marshall 213 Ed saturation is seen in ed-resolving ocean models. igher resolution ed saturation occurs Ed saturation was theoreticall predicted b Straub (1993) but with an entirel baroclinic argument. (vertical momentum transfer interfacial ed form stress) [There are man other eamples: allberg & Gnanadesikan 21, Tansle & Marshall 21, allberg & Gnanadesikan 26, ogg et al. 28, Nadeau & Straub 29, Farneti et al. 21, Nadeau & Straub 212, Meredith et al. 212, Morisson & ogg 213, Abernathe & Cessi 214, Farneti et al. 215, Nadeau & Ferrari 215, Marshall et al. 216.]
Question: So, does this barotropic QG model show ed saturation or not?
Ttw u u bot Ttw u Tbot let s use these two topographies p 2 cos (14/) T u rms ` q p tw 2 rms cos (14/).7 q 2 h i ` 2i hr hr 2 i TACC p u bot TACC 2 rms cos (14/) @ b Tbot 2 rms cos (1/) cos (1/) 2 rms cos (1/) cos (1/).7 q h 2 i u u u ` hr.7 tw2 i bot T tw u u tw u bot Ttw T bot u u tw u bot p 2 rms cos (14 1 the of Tbot TACC /9Ttw periodic TACC domain (both 2 rms cos (1/) cos (1/) h 2 i u bot @z u u bot Tbot Ttw 2 rms cos (1/) co 775 km 4 km 135 kg m 3 4 1 11 q 1 1 lat 6 S ) f 1.26 1 s, 1.14 1 m sh 2 i topographies impl the same length-scale: 2 i.7 ) 6 1 ` hr hrms 2 m ) rms 6.3 1 s µ 6.3 1 8 s 1 (18 das) 1 put some quasi-realistic numbers 775 km 4 km 135 kg m f & for 6 S 1 µ (18 das) hrms 2 m ) rms (1.8 das) 1 3 6 and var the 6wind stress TACC u Ttw
T bot how does the transport U var with wind stress? T tw ū bot ū p 2 rms cos (14/) ū bot transport ` q h 2 i hr 2 i.7 ū T ACC 2 rms cos (1/)cos(1/) ū tw ū bot T tw T bot 775km 4km 135kgm 3 lat 6 S ) f 1.26 1 4 s 1, 1.14 1 11 m ed 1 s 1 saturation h rms 2m) rms 6.3 1 6 s 1 µ 6.3 1 8 s 1 (18 das) 1 wind stress do we understand wh? wind stress.2nm 2, F U increases 4 times ` 2.2 rms over a 6-fold wind stress increase Inspired b the Southern Ocean we take 775km, 4km, 135kgm 3, p
T bot how does the transport U var with wind stress? T tw ū bot ū p 2 rms cos (14/) ū bot transport ` q h 2 i hr 2 i.7 ū T ACC 2 rms cos (1/)cos(1/) flat-bottom transport value ū tw ū bot T tw T bot 1 Sv 775km 4km 135kgm 3 lat 6 S ) f 1.26 1 4 s 1, 1.14 1 11 m ed 1 s 1 saturation h rms 2m) rms 6.3 1 6 s 1 µ 6.3 1 8 s 1 (18 das) 1 wind stress do we understand wh? wind stress.2nm 2, F U increases 4 times ` 2.2 rms over a 6-fold wind stress increase Inspired b the Southern Ocean we take 775km, 4km, 135kgm 3, p
Geostrophic contours geostrophic contours (, ). this is small-rossb number epansion of f/(h) The main control parameter for whether ed saturation occurs ofthe geostrophic contours. t is the structure z f v (@ u@ v@ w@ ) closed geostrophic contours (@t u@ v@ w@z ) u @ open geostrophic contours are closed f ( ) p z a ua most p @contours va geostrophic h f ( ) h () ( h) h p { all geostrophic contours are open
Geostrophic contours geostrophic contours (, ). this is small-rossb number epansion of f/(h) The main control parameter for whether ed saturation occurs ofthe geostrophic contours. t is the structure z f v (@ u@ v@ w@ ) closed geostrophic contours (@t u@ v@ w@z ) u @ open geostrophic contours are closed f ( ) p z this topograph ehibits ed saturation a ua most p @contours va geostrophic h f ( ) h () ( h) h p { all geostrophic contours are open
(, ) geostrophic contours. this is small-rossb number epansion of f/(h) Ed saturation occurs when the geostrophic contours are open, that is, when the geostrophic contours span the domain in the zonal direction. this is a general result we ve seen it in various cases whatever the topograph
further smptoms of ed saturation fied F wind stress EKE grows roughl linearl with wind stress bottom drag transport grows with increasing bottom drag [ogg & Blundell 26, Nadeau & Straub 212, Nadeau & Ferrari 215, Marshall et al. 216]
take home messages Ed saturation can occur without baroclinicit! The barotropic QG model shows ed saturation when geostrophic contours are open. This is surprising! All previous arguments were based on baroclinicit. This, does not preclude the role of baroclinic processes in the ACC equilibration. But it does argues that barotropic processes do contribute. We need new process models of baroclinic turbulence in which the mean flow is wind-driven and topograph eerts form stress. (In progress) thank ou Constantinou (217). A barotropic model of ed saturation. JPO (submitted, arxiv:173.6594) Constantinou and Young (217). Beta-plane turbulence above monoscale topograph. JFM (in revision, arxiv:1612.3374)