Parameterisation of meso-scale eddy mixing · 2015. 6. 14. · Parameterisation of meso-scale eddy mixing Carsten Eden IFM-GEOMAR, Kiel with Richard Greatbatch, Jürgen Willebrand,

Parameterisation of meso-scale eddy mixing

Carsten Eden IFM-GEOMAR, Kiel

with Richard Greatbatch, Jürgen Willebrand, Dirk Olbers

Exeter 28. April 2009 1

Parameterisation of meso-scale eddy mixing

Carsten Eden IFM-GEOMAR, Kiel

with Richard Greatbatch, Jürgen Willebrand, Dirk Olbers

. (FLAME) models of different horizontal grid resolution

. coarse ocean climate model→ ‘‘eddy-permitting’’ ocean model


Interpretation of mesoscale eddy mixing

. mean buoyancy (tracer) budget after Reynolds averaging

∂

∂tb+ u · ∇b+∇ · u′b′ = Q

. decompose eddy flux u′b′ in direction of ∇b and perpendicular, denoted by ∇¬ b

u′b′ = −K∇b+ B ∇¬b

Depth

Latitude

ρ=const



. mean buoyancy (tracer) budget after Reynolds averaging

∂

∂tb+ u · ∇b+∇ · u′b′ = Q


u′b′ = −K∇b+ B ∇¬b

. mean buoyancy budget becomes

bt + (u+ ∇¬B) · ∇b = ∇ ·K∇b+ Q




u′b′ = −K∇b+ B ∇¬b

. Transformed Eulerian Mean (TEM/TRM) formalism

bt + (u+ ∇¬B) · ∇b = ∇ ·K∇b+ Q

. K is a turbulent (diapycnal) diffusivity

. (isotropic) diffusive effect of mesoscale eddies

Depth

Latitude

ρ=const




u′b′ = −K∇b+ B ∇¬b

. Transformed Eulerian Mean (TEM/TRM) formalism

bt + (u+ ∇¬B) · ∇b = ∇ ·K∇b+ Q

. B is streamfunction for eddy-driven advection u∗ = ∇¬B

. ‘‘advective mixing’’ effect of mesoscale eddies

Depth

Latitude

ρ=const

B=const



bt + (u + u∗) · ∇b = ∇ ·K∇b+ Q

⇒ u∗ = ∇×B is

. Bolus velocity

. Quasi-Stokes drift

. Gent-McWilliams parameterisation

. eddy-induced advection



bt + (u + u∗) · ∇b = ∇ ·K∇b+ Q

⇒ u∗ = ∇×B is

. Bolus velocity

. Quasi-Stokes drift

. Gent-McWilliams parameterisation

. eddy-induced advection

⇒ u+ u∗ is

. residual velocity

. Transformed Eulerian Mean (TEM)

. Temporal Residual Mean (TRM)


interpretation of meso-scale eddy mixing

. Transformed Eulerian Mean (TEM) decomposition

u′b′ = −K∇b+B ×∇b+∇× θ

. (vector) streamfunction B = −|∇b|−2u′b′ ×∇b

B ≈ b−2z

0B@ bzv′b′

−bzu′b′u′b′ by − v′b′ bx

1CA for |bz| >> |∇hb|


interpretation of meso-scale eddy mixing

. Transformed Eulerian Mean (TEM) decomposition

u′b′ = −K∇b+B ×∇b+∇× θ

. (vector) streamfunction B = −|∇b|−2u′b′ ×∇b

B ≈ b−2z

0B@ bzv′b′

−bzu′b′u′b′ by − v′b′ bx

1CA for |bz| >> |∇hb|

. Gent-McWilliams parameterisation, isopycnal thickness diffusivity κ

u′hb′ ≈ −κ∇hb

. eddy-driven velocity u∗ = ∇×B =

0BB@−(κ bx

bz)z

−(κbybz

)z

(κ bxbz

)x + (κbybz

)y

1CCA


Some choices for thickness diffusivity κ

. constant value of κ = o(1000)m2/s




. exponentially decreasing with depth κ(z) ∼ ez/500m

(Danabasoglu + McWilliams, 1995)






. proportionally to vertically averaged horizontal stratification κ(x, y) ∼ |∇b|

(Griffies et al 2005)








. proportionally to local vertical stratification κ(x, y, z) ∼ bz

(Danabasoglu + Marshall, 2007)










. dimensional analysis κ(x, y) ∼ l2/τ with Eady growth rate τ and length scale l

(Visbeck et al 1997)










. dimensional analysis κ(x, y) ∼ l2/τ with Eady growth rate τ and length scale l

(Visbeck et al 1997)

. parameterisations for κ

(Killworth 1997/2001, Canuto + Dubovikov 2006, Eden + Greatbatch 2008)


Why care about better parameterisation for κ?

. decadal trend in zonal mean zonal wind stress over Southern Ocean

. decadal scale increase in MOC in Southern Ocean?





. anthropogenic carbon sink changes to source?

(Le Quere et al, 2007 Lovenduski et al, 2008)





. anthropogenic carbon sink changes to source?

(Le Quere et al, 2007 Lovenduski et al, 2008)

. natural CO2 outgassing vs.

anthropogenic CO2 uptake


The MOC in an idealised model of Southern Ocean

. zonally periodic boundary conditions

. west wind over Southern Ocean and ACC

. cooling in North (Atlantic) and South, warming inbetween


. zonal momentum budget ��ut − fv = ��−(u′v′)y + τz


. zonal momentum budget ��ut − fv = ��−(u′v′)y + τz

. vertical intregration from surface to depth z yields fM = −τo

with streamfunction ∇¬M = (−Mz,My) = (v, w) and surface wind stress τo


. Deacon cell in mean MOC streamfunctionM ≈ −τo/f

Olbers + Visbeck (2005), Radko + Marshall (2003)


. Deacon cell in mean MOC streamfunctionM ≈ −τo/f

. eddy driven MOC B ≈ −v′b′/bz = κby/bz ≡ κs


. Deacon cell in mean MOC streamfunctionM ≈ −τ/f

. eddy driven MOC B ≈ −v′b′/bz = κby/bz ≡ κs

. residual MOC ψ = κs− τ/f with ��bt + ∇¬ψ · ∇b = �

�Q, i.e. J(ψ, b) = 0


. 2,4 times increased wind stress τ o→ 2,4 times increased mean MOC,M = −τo/f

. isopycnal slopes s do not change much

. what about residual MOC κs− τ/f ?


. slopes s do not change much

. thickness diffusivity κ increases significantly

. residual MOC κs− τ/f increases significantly less than with constant K



. residual MOC κs− τ/f increases significantly less than with constant κ

. ocean climate model have constant κ

. climate models cannot predict climate change in Southern Ocean

. need of parameterisation for κ



. residual MOC κs− τ/f increases significantly less than with constant κ

. ocean climate model have constant κ

. climate models cannot predict climate change in Southern Ocean

. need of parameterisation for κ

. estimate κ from eddy-permitting models

. parameterise κ

. evaluate parameterised κ for mean state

and climate change in Southern Ocean


estimate κ from eddy-resolving models

. thickness diffusivity κ diagnosed in eddy resolving models


estimate κ from eddy-resolving models

. Vertical dependency of κ


how to parameterise thickness diffusivity κ?

. Visbeck et al (1997): from dimensional analysis κ ∼ l2/τ

with τ Eady growth rate and length scale l





. Killworth (1997/2001) found from linear theory

v∗ = κ(by/bz)z + κβ/f instead of TEM/GM form v∗ = (κby/bz)z







with κ ∼ koci| pu−c|2

where wavenumber ko at max. growth rates, phase speed c and eigenfunction p

from (quasi-geostrophic) eigenvalue problem for background flow u










. based on linearized equations

. complicated form of v∗

. local eigenvalue problem has to be solved










. based on linearized equations

. complicated form of v∗

. local eigenvalue problem has to be solved

. Canuto + Dubovikov (2005/2006) extended Killworth (1997/2001)

included also closure for nonlinear terms

found extra terms for v∗



. mixing length approach by Green (1970): κ = UeddyL



. mixing length approach by Green (1970): κ = UeddyL = e1/2L




. use diagnostic eddy length scale L





. but prognostic budget for eddy kinetic energy e

∂

∂te+ uh · ∇e+∇ ·M = S + b′w′ − ε






∂

∂te+ uh · ∇e+∇ ·M = S + b′w′ − ε

. production of e by

. baroclinic instability b′w′






∂

∂te+ uh · ∇e+∇ ·M = S + b′w′ − ε



. barotropic instability S






∂

∂te+ uh · ∇e+∇ ·M = S + b′w′ − ε




. dissipation of e by ε






∂

∂te+ uh · ∇e+∇ ·M = S + b′w′ − ε




. dissipation of e by ε

. radiation byM = u′e+ p′u′


Parameterising thickness diffusivity κ

. assume (locally) zero diapycnal mixing by mesoscale eddies

u′b′ · ∇b = 0 → b′w′ = −b−1z u

′hb′ · ∇hb




u′b′ · ∇b = 0 → b′w′ = −b−1z u

′hb′ · ∇hb

. follow GM u′hb′ = −κ∇hb

b′w′ = κ|∇hb|2

bz




u′b′ · ∇b = 0 → b′w′ = −b−1z u

′hb′ · ∇hb


b′w′ = κ|∇hb|2

bz

. dissipation, ε, e.g. after Kolmogorov ε ∼ e3/2/L, bottom friction, ...




u′b′ · ∇b = 0 → b′w′ = −b−1z u

′hb′ · ∇hb


b′w′ = κ|∇hb|2

bz

. dissipation, ε, e.g. after Kolmogorov ε ∼ e3/2/L, bottom friction, ...

. radiation simply asM = −κ∇he



. simplest turbulence closure model

d

dte = S + b′w′ − ε−∇ ·M

. thickness (GM) diffusivity κ = e1/2L




d

dte = S + b′w′ − ε−∇ ·M


. eddy length scale L?




d

dte = S + b′w′ − ε−∇ ·M



. Rossby radius Lr




d

dte = S + b′w′ − ε−∇ ·M



. Rossby radius Lr

. Rhines scale LRhi =q

Uβ

compare turbulent velocity with Rossby wave speed

ω

k∼ Ueddy ∼

˛−β

k2 + L−2r

˛or L−2 ∼ L−2

Rhi + L−2r




d

dte = S + b′w′ − ε−∇ ·M



. Rossby radius Lr

. Rhines scale LRhi =q

Uβ

compare turbulent velocity with Rossby wave speed

ω

k∼ Ueddy ∼

˛−β

k2 + L−2r

˛or L−2 ∼ L−2

Rhi + L−2r

. take L = min(Lr, LRhi)


Length scale(s) in the North Atlantic

00 5050 100100 15015000

5050

100100

150150

200200

JASON

Model

L vs. min(Rossby, Rhines scale)L vs. Rossby radius

Eden (2007)

. estimated L fits better withmin(Lr, LRhi)

. consistent with Theiss (2004), Scott and Polvani (2007), Tulloch et al (2008)




d

dte = κ

|∇hb|2

bz−e3/2

L+∇h · κ∇e+ S

L = min (LRossby, LRhines)




d

dte = κ

|∇hb|2

bz−e3/2

L+∇h · κ∇e+ S


. assume local balance and neglect barotropic instability S

κ = σL2, L = min (Lr, σ/β)

where σ = f |uz|N−1 is the Eady growth rate




d

dte = κ

|∇hb|2

bz−e3/2

L+∇h · κ∇e+ S


. assume local balance and neglect barotropic instability S

κ = σL2, L = min (Lr, σ/β)

where σ = f |uz|N−1 is the Eady growth rate

. similar to Visbeck et al (1997)

scaling as in Larichev and Held (1995) and Held and Larichev (1995)


Evaluation in eddy-resolving models

. κ at 300m depth inm2/s diagnosed from eddy-resolving model

. mixing length assumption κ = e1/2L

using e and L = min(LRossby, LRhines) from same model


Evaluation in coarse resolution models

. EKE at 300 m depth in log(e/[cm2/s])


Evaluation in coarse resolution models

. EKE at 300 m depth in log(e/[cm2/s])


Evaluation in global ocean model (CCSM4)

. simple local closure κ = σL2 vs Visbeck et al parameterisation



. simplest turbulence closure model: κ = e1/2L with

d

dte = κ

|∇hb|2

bz−e3/2

L+∇h · κ∇e+ S





d

dte = κ

|∇hb|2

bz−e3/2

L+∇h · κ∇e+ S


. κ = e1/2L works reasonable well in eddy-resolving models




d

dte = κ

|∇hb|2

bz−e3/2

L+∇h · κ∇e+ S



. first implementation of closure works surprisingly well




d

dte = κ

|∇hb|2

bz−e3/2

L+∇h · κ∇e+ S




. κ can be used as thickness diffusivity for GM and as isopycnal diffusivity




d

dte = κ

|∇hb|2

bz−e3/2

L+∇h · κ∇e+ S





. compare simulation of changes in Southern Ocean with different κ




d

dte = κ

|∇hb|2

bz−e3/2

L+∇h · κ∇e+ S





. compare simulation of changes in Southern Ocean with different κ

. mean momentum budget?


Wide channel model

. revisit eddy closure of Eden+Greatbatch (2008) in idealised model

. effects eddy momentum fluxes

2000 km

4000 km

2000 km

0 km

0 km 4000 km

0.05

0.04

0.03

0.02

0.01

0

. u and b at 500m depth

. zonally reentrant wide channel

. 30km× 30km horizontal resolution

40 levels with 50m

. flat bottom, β plane

. forcing by restoring zones at northernand southern wall

. dissipation by interior drag and bihar-monic friction and mixing


Wide channel model

0

250

500

750

1000

0km 4000km2000km0km 2000km 4000km

0

0.8

0.4

−0.4

−0.8−2000m

−1000m

0mEKE and EKE productionMean flow and buoyancy

. eastward zonal jets

. baroclinic instability in the interior

. eddy momentum fluxes driving zonal jets


Closure for zonally averaged model

. QG scaling, zonally averaged equations

ut = fv − (v′u′)y + fric , bt = −w − (v′b′)y + diff

with eddy buoyancy flux v′b′, and eddy momentum flux v′u′

(b = b∗/N2 denotes scaled buoyancy b∗)







. parameterise v′b′ = −κby with thickness diffusivity κ








. closure for eddy momentum flux v′u′?








. closure for eddy momentum flux v′u′?

. assume PV mixing


Eddy closure: PV budget, gauge term

. Zonally averaged budget for PV, q = −uy + fbz + βy

qt = −(v′q′)y + fric+ diff

with eddy PV fluxes v′q′






. use equivalence between eddy PV and eddy buoyancy and momentum fluxes

v′q′ = −(v′u′)y + f(v′b′)z + θ

with gauge function θ(z, t)!







v′q′ = −(v′u′)y + f(v′b′)z + θ


. assume PV mixing with same diffusivity as buoyancy, i.e. v′q′ = −κqy







v′q′ = −(v′u′)y + f(v′b′)z + θ



. and get (v′u′)y = f(v′b′)z − v′q′ + θ







v′q′ = −(v′u′)y + f(v′b′)z + θ



. and get (v′u′)y = f(v′b′)z − v′q′ + θ = −κuyy − κzfby + κβ + θ







v′q′ = −(v′u′)y + f(v′b′)z + θ



. and get (v′u′)y = f(v′b′)z − v′q′ + θ = −κuyy − κzfby + κβ + θ

. mean momentum budget becomes

ut = fv + κ(uyy − β) + κzfby − θ + fric


. Diffusivity estimated from v′b′ = −κby and v′q′ = −κqy in log10(κ/[m2/s])


Eddy closure

. parameterised zonally averaged model

bt = −w − (κby)y + diff



Eddy closure




. Eden + Greatbatch (2008) suggest:

adjust θ to satisfy global momentum constrain of Bretherton (1966)

i.e. no mean force by eddies, just redistribution of momentum

−Z 0

−h

Z L

0

sydydz =

Z 0

−h

Z L

0

(κ(uyy − β) + κzfby − θ)dydz = 0


Eddy closure







−Z 0

−h

Z L

0

sydydz =

Z 0

−h

Z L

0


. in contrast to Wardle + Marshall (2000) or Olbers et al (2000) who suggest:

adjust diffusivity κ to satisfy global momentum constrain (and set θ = 0)


Eddy closure







−Z 0

−h

Z L

0

sydydz =

Z 0

−h

Z L

0


. in contrast to Wardle + Marshall (2000) or Olbers et al (2000) who suggest:

adjust diffusivity κ to satisfy global momentum constrain (and set θ = 0)

. freedom to use closure for κ, e.g. as given by Eden + Greatbatch (2008)


zonally averaged model

0 2000 4000

Eddy resolving model K= const

. Zonal mean zonal flow u inm/s

in eddy resolving model and zonally averaged (parameterised) model

. for constant κ→ no jets

. inhomogenity in κ produces jets


. Diffusivity κ and u (contours)


Eddy closure: prescribed diffusivity


bt = −w − (κby)y + Q

ut = fv + κ(uyy − β) + κzfby − θ

−Z 0

−h

Z L

0

sydydz =

Z 0

−h

Z L

0







−Z 0

−h

Z L

0

sydydz =

Z 0

−h

Z L

0


. consider prescribed diffusivity, e.g. κ = 5000m2/s (1 + 0.1 sin 8πy/L)

to mimick minima of κ within zonal jets






−Z 0

−h

Z L

0

sydydz =

Z 0

−h

Z L

0




. gauge function becomes θ = 1/LR L

0(κ(uyy − β)dy ≈ −β/L

R L0Kdy






−Z 0

−h

Z L

0

sydydz =

Z 0

−h

Z L

0






R L0Kdy ≡ κ






−Z 0

−h

Z L

0

sydydz =

Z 0

−h

Z L

0






R L0Kdy ≡ κ

ut = fv + κuyy + β(κ− κ)






−Z 0

−h

Z L

0

sydydz =

Z 0

−h

Z L

0






R L0Kdy ≡ κ


dominant balance in momentum budget is κuyy + β(κ− κ) ≈ 0






−Z 0

−h

Z L

0

sydydz =

Z 0

−h

Z L

0






R L0Kdy ≡ κ


dominant balance in momentum budget is κuyy + β(κ− κ) ≈ 0

. local minimum in κ means κuyy > 0 in momentum balance→ eastward jet



0 2000 4000 0 2000 4000

Eddy resolving model K = 5000 (1+0.1 sin(8piL/y))





0 2000 4000 0 2000 4000

Eddy resolving model K = 5000 (1+0.1 sin(8piL/y))



. prescribed local minima in κ produce zonal jets

as a consequence of PV mixing and global momentum constraint


flow interactive diffusivity

2000 40000

param. zonal mean flow param. EKE and EKE production



. using closure of Eden + Greatbatch (2008) with fixed eddy length scale

. minima of κ, eastward jets


flow interactive diffusivity

2000 40000

param. zonal mean flow param. EKE and EKE production



. using closure of Eden + Greatbatch (2008) with fixed eddy length scale

. minima of κ, eastward jets

. but surface minima in EKE in jets since κ = e1/2L

. prognostic eddy length scale ... ?


Conclusions

. need for a better parameterisation of thickness diffusivity κ

such that climate models can predict climate change in Southern Ocean


Conclusions



. implementation of simple closure for κ based on mixing length assumption

in buoyancy budget works reasonable well


Conclusions





. test of parameterisations of κ for climate change in Southern Ocean


Conclusions






. meso-scale eddy closure based on buoyancy and PV mixing


Conclusions







. closure for momentum budget


Conclusions








. gauge function to satisfy global momentum constraint


Conclusions









. minima in κ produce eastward jets


Conclusions









. minima in κ produce eastward jets

. κ = e1/2L with fixed L produce jets

but fails to produce EKE correctly

higher order closure?


Documents

Parameterisation of meso-scale eddy mixing · 2015. 6. 14. · Parameterisation of meso-scale eddy mixing Carsten Eden IFM-GEOMAR, Kiel with Richard Greatbatch, Jürgen Willebrand,