Post on 31-Mar-2021
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
Exeter 28. April 2009 1
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
Exeter 28. April 2009 2
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
. mean buoyancy budget becomes
bt + (u+ ∇¬B) · ∇b = ∇ ·K∇b+ Q
Exeter 28. April 2009 3
Interpretation of mesoscale eddy mixing
. decompose eddy flux u′b′ in direction of ∇b and perpendicular, denoted by ∇¬ b
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
Exeter 28. April 2009 4
Interpretation of mesoscale eddy mixing
. decompose eddy flux u′b′ in direction of ∇b and perpendicular, denoted by ∇¬ b
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
Exeter 28. April 2009 5
Interpretation of mesoscale eddy mixing
bt + (u + u∗) · ∇b = ∇ ·K∇b+ Q
⇒ u∗ = ∇×B is
. Bolus velocity
. Quasi-Stokes drift
. Gent-McWilliams parameterisation
. eddy-induced advection
Exeter 28. April 2009 6
Interpretation of mesoscale eddy mixing
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)
Exeter 28. April 2009 6
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|
Exeter 28. April 2009 7
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
Exeter 28. April 2009 7
Some choices for thickness diffusivity κ
. constant value of κ = o(1000)m2/s
Exeter 28. April 2009 8
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)
Exeter 28. April 2009 8
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)
Exeter 28. April 2009 8
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)
Exeter 28. April 2009 8
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)
Exeter 28. April 2009 8
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)
. parameterisations for κ
(Killworth 1997/2001, Canuto + Dubovikov 2006, Eden + Greatbatch 2008)
Exeter 28. April 2009 8
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?
Exeter 28. April 2009 9
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)
Exeter 28. April 2009 9
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)
. natural CO2 outgassing vs.
anthropogenic CO2 uptake
Exeter 28. April 2009 9
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
Exeter 28. April 2009 10
. zonal momentum budget ��ut − fv = ��������−(u′v′)y + τz
Exeter 28. April 2009 11
. 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
Exeter 28. April 2009 11
. Deacon cell in mean MOC streamfunctionM ≈ −τo/f
Olbers + Visbeck (2005), Radko + Marshall (2003)
Exeter 28. April 2009 12
. Deacon cell in mean MOC streamfunctionM ≈ −τo/f
. eddy driven MOC B ≈ −v′b′/bz = κby/bz ≡ κs
Exeter 28. April 2009 13
. 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
Exeter 28. April 2009 14
. 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 ?
Exeter 28. April 2009 15
. slopes s do not change much
. thickness diffusivity κ increases significantly
. residual MOC κs− τ/f increases significantly less than with constant K
Exeter 28. April 2009 16
Why care about better 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
Exeter 28. April 2009 17
Why care about better 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
Exeter 28. April 2009 17
estimate κ from eddy-resolving models
. thickness diffusivity κ diagnosed in eddy resolving models
Exeter 28. April 2009 18
estimate κ from eddy-resolving models
. Vertical dependency of κ
Exeter 28. April 2009 19
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
Exeter 28. April 2009 20
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
Exeter 28. April 2009 20
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
Exeter 28. April 2009 20
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
Exeter 28. April 2009 20
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
. Canuto + Dubovikov (2005/2006) extended Killworth (1997/2001)
included also closure for nonlinear terms
found extra terms for v∗
Exeter 28. April 2009 20
how to parameterise thickness diffusivity κ?
. mixing length approach by Green (1970): κ = UeddyL
Exeter 28. April 2009 21
how to parameterise thickness diffusivity κ?
. mixing length approach by Green (1970): κ = UeddyL = e1/2L
Exeter 28. April 2009 21
how to parameterise thickness diffusivity κ?
. mixing length approach by Green (1970): κ = UeddyL = e1/2L
. use diagnostic eddy length scale L
Exeter 28. April 2009 21
how to parameterise thickness diffusivity κ?
. 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
Exeter 28. April 2009 21
how to parameterise thickness diffusivity κ?
. 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
. production of e by
. baroclinic instability b′w′
Exeter 28. April 2009 21
how to parameterise thickness diffusivity κ?
. 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
. production of e by
. baroclinic instability b′w′
. barotropic instability S
Exeter 28. April 2009 21
how to parameterise thickness diffusivity κ?
. 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
. production of e by
. baroclinic instability b′w′
. barotropic instability S
. dissipation of e by ε
Exeter 28. April 2009 21
how to parameterise thickness diffusivity κ?
. 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
. production of e by
. baroclinic instability b′w′
. barotropic instability S
. dissipation of e by ε
. radiation byM = u′e+ p′u′
Exeter 28. April 2009 21
Parameterising thickness diffusivity κ
. assume (locally) zero diapycnal mixing by mesoscale eddies
u′b′ · ∇b = 0 → b′w′ = −b−1z u
′hb′ · ∇hb
Exeter 28. April 2009 22
Parameterising thickness diffusivity κ
. assume (locally) zero diapycnal mixing by mesoscale eddies
u′b′ · ∇b = 0 → b′w′ = −b−1z u
′hb′ · ∇hb
. follow GM u′hb′ = −κ∇hb
b′w′ = κ|∇hb|2
bz
Exeter 28. April 2009 22
Parameterising thickness diffusivity κ
. assume (locally) zero diapycnal mixing by mesoscale eddies
u′b′ · ∇b = 0 → b′w′ = −b−1z u
′hb′ · ∇hb
. follow GM u′hb′ = −κ∇hb
b′w′ = κ|∇hb|2
bz
. dissipation, ε, e.g. after Kolmogorov ε ∼ e3/2/L, bottom friction, ...
Exeter 28. April 2009 22
Parameterising thickness diffusivity κ
. assume (locally) zero diapycnal mixing by mesoscale eddies
u′b′ · ∇b = 0 → b′w′ = −b−1z u
′hb′ · ∇hb
. follow GM u′hb′ = −κ∇hb
b′w′ = κ|∇hb|2
bz
. dissipation, ε, e.g. after Kolmogorov ε ∼ e3/2/L, bottom friction, ...
. radiation simply asM = −κ∇he
Exeter 28. April 2009 22
Parameterising thickness diffusivity κ
. simplest turbulence closure model
d
dte = S + b′w′ − ε−∇ ·M
. thickness (GM) diffusivity κ = e1/2L
Exeter 28. April 2009 23
Parameterising thickness diffusivity κ
. simplest turbulence closure model
d
dte = S + b′w′ − ε−∇ ·M
. thickness (GM) diffusivity κ = e1/2L
. eddy length scale L?
Exeter 28. April 2009 23
Parameterising thickness diffusivity κ
. simplest turbulence closure model
d
dte = S + b′w′ − ε−∇ ·M
. thickness (GM) diffusivity κ = e1/2L
. eddy length scale L?
. Rossby radius Lr
Exeter 28. April 2009 23
Parameterising thickness diffusivity κ
. simplest turbulence closure model
d
dte = S + b′w′ − ε−∇ ·M
. thickness (GM) diffusivity κ = e1/2L
. eddy length scale L?
. 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
Exeter 28. April 2009 23
Parameterising thickness diffusivity κ
. simplest turbulence closure model
d
dte = S + b′w′ − ε−∇ ·M
. thickness (GM) diffusivity κ = e1/2L
. eddy length scale L?
. 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)
Exeter 28. April 2009 23
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)
Exeter 28. April 2009 24
Parameterising thickness diffusivity κ
. simplest turbulence closure model
d
dte = κ
|∇hb|2
bz−e3/2
L+∇h · κ∇e+ S
L = min (LRossby, LRhines)
Exeter 28. April 2009 25
Parameterising thickness diffusivity κ
. simplest turbulence closure model
d
dte = κ
|∇hb|2
bz−e3/2
L+∇h · κ∇e+ S
L = min (LRossby, LRhines)
. 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
Exeter 28. April 2009 25
Parameterising thickness diffusivity κ
. simplest turbulence closure model
d
dte = κ
|∇hb|2
bz−e3/2
L+∇h · κ∇e+ S
L = min (LRossby, LRhines)
. 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)
Exeter 28. April 2009 25
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
Exeter 28. April 2009 26
Evaluation in coarse resolution models
. EKE at 300 m depth in log(e/[cm2/s])
Exeter 28. April 2009 27
Evaluation in coarse resolution models
. EKE at 300 m depth in log(e/[cm2/s])
Exeter 28. April 2009 28
Evaluation in global ocean model (CCSM4)
. simple local closure κ = σL2 vs Visbeck et al parameterisation
Exeter 28. April 2009 29
Parameterising thickness diffusivity κ
. simplest turbulence closure model: κ = e1/2L with
d
dte = κ
|∇hb|2
bz−e3/2
L+∇h · κ∇e+ S
L = min (LRossby, LRhines)
Exeter 28. April 2009 30
Parameterising thickness diffusivity κ
. simplest turbulence closure model: κ = e1/2L with
d
dte = κ
|∇hb|2
bz−e3/2
L+∇h · κ∇e+ S
L = min (LRossby, LRhines)
. κ = e1/2L works reasonable well in eddy-resolving models
Exeter 28. April 2009 30
Parameterising thickness diffusivity κ
. simplest turbulence closure model: κ = e1/2L with
d
dte = κ
|∇hb|2
bz−e3/2
L+∇h · κ∇e+ S
L = min (LRossby, LRhines)
. κ = e1/2L works reasonable well in eddy-resolving models
. first implementation of closure works surprisingly well
Exeter 28. April 2009 30
Parameterising thickness diffusivity κ
. simplest turbulence closure model: κ = e1/2L with
d
dte = κ
|∇hb|2
bz−e3/2
L+∇h · κ∇e+ S
L = min (LRossby, LRhines)
. κ = e1/2L works reasonable well in eddy-resolving models
. first implementation of closure works surprisingly well
. κ can be used as thickness diffusivity for GM and as isopycnal diffusivity
Exeter 28. April 2009 30
Parameterising thickness diffusivity κ
. simplest turbulence closure model: κ = e1/2L with
d
dte = κ
|∇hb|2
bz−e3/2
L+∇h · κ∇e+ S
L = min (LRossby, LRhines)
. κ = e1/2L works reasonable well in eddy-resolving models
. first implementation of closure works surprisingly well
. κ can be used as thickness diffusivity for GM and as isopycnal diffusivity
. compare simulation of changes in Southern Ocean with different κ
Exeter 28. April 2009 30
Parameterising thickness diffusivity κ
. simplest turbulence closure model: κ = e1/2L with
d
dte = κ
|∇hb|2
bz−e3/2
L+∇h · κ∇e+ S
L = min (LRossby, LRhines)
. κ = e1/2L works reasonable well in eddy-resolving models
. first implementation of closure works surprisingly well
. κ can be used as thickness diffusivity for GM and as isopycnal diffusivity
. compare simulation of changes in Southern Ocean with different κ
. mean momentum budget?
Exeter 28. April 2009 30
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
Exeter 28. April 2009 31
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
Exeter 28. April 2009 32
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∗)
Exeter 28. April 2009 33
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
Exeter 28. April 2009 33
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′?
Exeter 28. April 2009 33
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′?
. assume PV mixing
Exeter 28. April 2009 33
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′
Exeter 28. April 2009 34
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)!
Exeter 28. April 2009 34
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)!
. assume PV mixing with same diffusivity as buoyancy, i.e. v′q′ = −κqy
Exeter 28. April 2009 34
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)!
. assume PV mixing with same diffusivity as buoyancy, i.e. v′q′ = −κqy
. and get (v′u′)y = f(v′b′)z − v′q′ + θ
Exeter 28. April 2009 34
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)!
. assume PV mixing with same diffusivity as buoyancy, i.e. v′q′ = −κqy
. and get (v′u′)y = f(v′b′)z − v′q′ + θ = −κuyy − κzfby + κβ + θ
Exeter 28. April 2009 34
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)!
. assume PV mixing with same diffusivity as buoyancy, i.e. v′q′ = −κqy
. 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
Exeter 28. April 2009 34
. Diffusivity estimated from v′b′ = −κby and v′q′ = −κqy in log10(κ/[m2/s])
Exeter 28. April 2009 35
Eddy closure
. parameterised zonally averaged model
bt = −w − (κby)y + diff
ut = fv + κ(uyy − β) + κzfby − θ + fric
Exeter 28. April 2009 36
Eddy closure
. parameterised zonally averaged model
bt = −w − (κby)y + diff
ut = fv + κ(uyy − β) + κzfby − θ + fric
. 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
Exeter 28. April 2009 36
Eddy closure
. parameterised zonally averaged model
bt = −w − (κby)y + diff
ut = fv + κ(uyy − β) + κzfby − θ + fric
. 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
. 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)
Exeter 28. April 2009 36
Eddy closure
. parameterised zonally averaged model
bt = −w − (κby)y + diff
ut = fv + κ(uyy − β) + κzfby − θ + fric
. 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
. 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)
Exeter 28. April 2009 36
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
Exeter 28. April 2009 37
. Diffusivity κ and u (contours)
Exeter 28. April 2009 38
Eddy closure: prescribed diffusivity
. parameterised zonally averaged model
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
(κ(uyy − β) + κzfby − θ)dydz = 0
Exeter 28. April 2009 39
Eddy closure: prescribed diffusivity
. parameterised zonally averaged model
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
(κ(uyy − β) + κzfby − θ)dydz = 0
. consider prescribed diffusivity, e.g. κ = 5000m2/s (1 + 0.1 sin 8πy/L)
to mimick minima of κ within zonal jets
Exeter 28. April 2009 39
Eddy closure: prescribed diffusivity
. parameterised zonally averaged model
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
(κ(uyy − β) + κzfby − θ)dydz = 0
. consider prescribed diffusivity, e.g. κ = 5000m2/s (1 + 0.1 sin 8πy/L)
to mimick minima of κ within zonal jets
. gauge function becomes θ = 1/LR L
0(κ(uyy − β)dy ≈ −β/L
R L0Kdy
Exeter 28. April 2009 39
Eddy closure: prescribed diffusivity
. parameterised zonally averaged model
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
(κ(uyy − β) + κzfby − θ)dydz = 0
. consider prescribed diffusivity, e.g. κ = 5000m2/s (1 + 0.1 sin 8πy/L)
to mimick minima of κ within zonal jets
. gauge function becomes θ = 1/LR L
0(κ(uyy − β)dy ≈ −β/L
R L0Kdy ≡ κ
Exeter 28. April 2009 39
Eddy closure: prescribed diffusivity
. parameterised zonally averaged model
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
(κ(uyy − β) + κzfby − θ)dydz = 0
. consider prescribed diffusivity, e.g. κ = 5000m2/s (1 + 0.1 sin 8πy/L)
to mimick minima of κ within zonal jets
. gauge function becomes θ = 1/LR L
0(κ(uyy − β)dy ≈ −β/L
R L0Kdy ≡ κ
ut = fv + κuyy + β(κ− κ)
Exeter 28. April 2009 39
Eddy closure: prescribed diffusivity
. parameterised zonally averaged model
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
(κ(uyy − β) + κzfby − θ)dydz = 0
. consider prescribed diffusivity, e.g. κ = 5000m2/s (1 + 0.1 sin 8πy/L)
to mimick minima of κ within zonal jets
. gauge function becomes θ = 1/LR L
0(κ(uyy − β)dy ≈ −β/L
R L0Kdy ≡ κ
ut = fv + κuyy + β(κ− κ)
dominant balance in momentum budget is κuyy + β(κ− κ) ≈ 0
Exeter 28. April 2009 39
Eddy closure: prescribed diffusivity
. parameterised zonally averaged model
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
(κ(uyy − β) + κzfby − θ)dydz = 0
. consider prescribed diffusivity, e.g. κ = 5000m2/s (1 + 0.1 sin 8πy/L)
to mimick minima of κ within zonal jets
. gauge function becomes θ = 1/LR L
0(κ(uyy − β)dy ≈ −β/L
R L0Kdy ≡ κ
ut = fv + κuyy + β(κ− κ)
dominant balance in momentum budget is κuyy + β(κ− κ) ≈ 0
. local minimum in κ means κuyy > 0 in momentum balance→ eastward jet
Exeter 28. April 2009 39
zonally averaged model
0 2000 4000 0 2000 4000
Eddy resolving model K = 5000 (1+0.1 sin(8piL/y))
. Zonal mean zonal flow u inm/s
in eddy resolving model and zonally averaged (parameterised) model
Exeter 28. April 2009 40
zonally averaged model
0 2000 4000 0 2000 4000
Eddy resolving model K = 5000 (1+0.1 sin(8piL/y))
. Zonal mean zonal flow u inm/s
in eddy resolving model and zonally averaged (parameterised) model
. prescribed local minima in κ produce zonal jets
as a consequence of PV mixing and global momentum constraint
Exeter 28. April 2009 40
flow interactive diffusivity
2000 40000
param. zonal mean flow param. EKE and EKE production
. Zonal mean zonal flow u inm/s
in eddy resolving model and zonally averaged (parameterised) model
. using closure of Eden + Greatbatch (2008) with fixed eddy length scale
. minima of κ, eastward jets
Exeter 28. April 2009 41
flow interactive diffusivity
2000 40000
param. zonal mean flow param. EKE and EKE production
. Zonal mean zonal flow u inm/s
in eddy resolving model and zonally averaged (parameterised) model
. 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 ... ?
Exeter 28. April 2009 41
Conclusions
. need for a better parameterisation of thickness diffusivity κ
such that climate models can predict climate change in Southern Ocean
Exeter 28. April 2009 42
Conclusions
. need for a better parameterisation of thickness diffusivity κ
such that climate models can predict climate change in Southern Ocean
. implementation of simple closure for κ based on mixing length assumption
in buoyancy budget works reasonable well
Exeter 28. April 2009 42
Conclusions
. need for a better parameterisation of thickness diffusivity κ
such that climate models can predict climate change in Southern Ocean
. implementation of simple closure for κ based on mixing length assumption
in buoyancy budget works reasonable well
. test of parameterisations of κ for climate change in Southern Ocean
Exeter 28. April 2009 42
Conclusions
. need for a better parameterisation of thickness diffusivity κ
such that climate models can predict climate change in Southern Ocean
. implementation of simple closure for κ based on mixing length assumption
in buoyancy budget works reasonable well
. test of parameterisations of κ for climate change in Southern Ocean
. meso-scale eddy closure based on buoyancy and PV mixing
Exeter 28. April 2009 42
Conclusions
. need for a better parameterisation of thickness diffusivity κ
such that climate models can predict climate change in Southern Ocean
. implementation of simple closure for κ based on mixing length assumption
in buoyancy budget works reasonable well
. test of parameterisations of κ for climate change in Southern Ocean
. meso-scale eddy closure based on buoyancy and PV mixing
. closure for momentum budget
Exeter 28. April 2009 42
Conclusions
. need for a better parameterisation of thickness diffusivity κ
such that climate models can predict climate change in Southern Ocean
. implementation of simple closure for κ based on mixing length assumption
in buoyancy budget works reasonable well
. test of parameterisations of κ for climate change in Southern Ocean
. meso-scale eddy closure based on buoyancy and PV mixing
. closure for momentum budget
. gauge function to satisfy global momentum constraint
Exeter 28. April 2009 42
Conclusions
. need for a better parameterisation of thickness diffusivity κ
such that climate models can predict climate change in Southern Ocean
. implementation of simple closure for κ based on mixing length assumption
in buoyancy budget works reasonable well
. test of parameterisations of κ for climate change in Southern Ocean
. meso-scale eddy closure based on buoyancy and PV mixing
. closure for momentum budget
. gauge function to satisfy global momentum constraint
. minima in κ produce eastward jets
Exeter 28. April 2009 42
Conclusions
. need for a better parameterisation of thickness diffusivity κ
such that climate models can predict climate change in Southern Ocean
. implementation of simple closure for κ based on mixing length assumption
in buoyancy budget works reasonable well
. test of parameterisations of κ for climate change in Southern Ocean
. meso-scale eddy closure based on buoyancy and PV mixing
. closure for momentum budget
. gauge function to satisfy global momentum constraint
. minima in κ produce eastward jets
. κ = e1/2L with fixed L produce jets
but fails to produce EKE correctly
higher order closure?
Exeter 28. April 2009 42