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Understanding physics-dynamics coupling in weather and climate...
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Understanding physics-dynamics coupling in
weather and climate models
Bob Beare
1and Mike Cullen
2
1University of Exeter.
2Met O�ce, Exeter.
7 July 2015
R. J. Beare
(a)400 m grid length
-4800 -2400 0 2400 4800x (m) x (m)
-4800
-2400
0
2400
4800
y (m)
-4800
-2400
0
2400
4800
y (m)
-5.5 -4.5 -3.5 -2.5 -1.5 -0.5 0.5 1.5 2.5 3.5 4.5 5.5
Vertical velocity (m s-1) Vertical velocity (m s-1)
Vertical velocity (m s-1) Vertical velocity (m s-1)
(b)200 m grid length
-4800 -2400 0 2400 4800
-4800 -2400 0 2400 4800x (m) x (m)
-4800 -2400 0 2400 4800
-4800
-2400
0
2400
4800
y (m)
-4800
-2400
0
2400
4800
y (m)
-5.5 -4.5 -3.5 -2.5 -1.5 -0.5 0.5 1.5 2.5 3.5 4.5 5.5
(c)100 m grid length
-5.5 -4.5 -3.5 -2.5 -1.5 -0.5 0.5 1.5 2.5 3.5
(d)50 m grid length
-5.5 -4.5 -3.5 -2.5 -1.5 -0.5 0.5 1.5 2.5 3.54.5 5.5 4.5 5.5
Fig. 2 Horizontal cross-sections of vertical velocity at height 500 m at time 3.5 h for horizontal grid lengthsof: a 400 m, b 200 m, c 100 m and d 50 m. Contour interval 1 m s−1
grid length of 400 m, the subgrid flux has become a significant proportion (46 %) of theresolved flux, confirming that the simulation is in the grey zone at this height.
At the inversion, however, the total buoyancy flux is not constant with resolution (Fig. 3).As the horizontal grid length is decreased, the entrainment flux decreases, although notmonotonically, to the converged value at 25 m, in agreement with Sullivan and Patton (2011).Whilst the total flux is relatively insensitive to resolution in the middle of the boundary layer,the resolved fields are likely to be more sensitive. It is the resolved fields that define whetherthe simulations are in the grey zone and are the focus of of our study.
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Introduction
IWeather and climate models partitioned into resolveddynamics and parametrized sub-grid processes (‘physics’).
IUnderstanding coupling of dynamics and physics central to
improving weather and climate models e.g. tropical
performance and diurnal cycle over land.
IComplex models should converge to a state of geostrophic
balance at small Rossby number (Ro). Convergence rate Ro
2,
Cullen (2008).
IDoes this idea extend to the physics? For boundary-layer
parametrization, use Ekman balance.
IUnphysical deviation from Ekman balance could be used to
diagnose numerical coupling problems.
ITest this approach for an idealised baroclinic wave.
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Introduction
IWeather and climate models partitioned into resolveddynamics and parametrized sub-grid processes (‘physics’).
IUnderstanding coupling of dynamics and physics central to
improving weather and climate models e.g. tropical
performance and diurnal cycle over land.
IComplex models should converge to a state of geostrophic
balance at small Rossby number (Ro). Convergence rate Ro
2,
Cullen (2008).
IDoes this idea extend to the physics? For boundary-layer
parametrization, use Ekman balance.
IUnphysical deviation from Ekman balance could be used to
diagnose numerical coupling problems.
ITest this approach for an idealised baroclinic wave.
2 / 13
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Introduction
IWeather and climate models partitioned into resolveddynamics and parametrized sub-grid processes (‘physics’).
IUnderstanding coupling of dynamics and physics central to
improving weather and climate models e.g. tropical
performance and diurnal cycle over land.
IComplex models should converge to a state of geostrophic
balance at small Rossby number (Ro). Convergence rate Ro
2,
Cullen (2008).
IDoes this idea extend to the physics? For boundary-layer
parametrization, use Ekman balance.
IUnphysical deviation from Ekman balance could be used to
diagnose numerical coupling problems.
ITest this approach for an idealised baroclinic wave.
2 / 13
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Introduction
IWeather and climate models partitioned into resolveddynamics and parametrized sub-grid processes (‘physics’).
IUnderstanding coupling of dynamics and physics central to
improving weather and climate models e.g. tropical
performance and diurnal cycle over land.
IComplex models should converge to a state of geostrophic
balance at small Rossby number (Ro). Convergence rate Ro
2,
Cullen (2008).
IDoes this idea extend to the physics? For boundary-layer
parametrization, use Ekman balance.
IUnphysical deviation from Ekman balance could be used to
diagnose numerical coupling problems.
ITest this approach for an idealised baroclinic wave.
2 / 13
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Introduction
IWeather and climate models partitioned into resolveddynamics and parametrized sub-grid processes (‘physics’).
IUnderstanding coupling of dynamics and physics central to
improving weather and climate models e.g. tropical
performance and diurnal cycle over land.
IComplex models should converge to a state of geostrophic
balance at small Rossby number (Ro). Convergence rate Ro
2,
Cullen (2008).
IDoes this idea extend to the physics? For boundary-layer
parametrization, use Ekman balance.
IUnphysical deviation from Ekman balance could be used to
diagnose numerical coupling problems.
ITest this approach for an idealised baroclinic wave.
2 / 13
![Page 7: Understanding physics-dynamics coupling in weather and climate …emps.exeter.ac.uk/media/universityofexeter/emps/climathnet/Robert... · I Understanding coupling of dynamics and](https://reader033.fdocuments.net/reader033/viewer/2022042401/5f0fe40f7e708231d44665dd/html5/thumbnails/7.jpg)
Introduction
IWeather and climate models partitioned into resolveddynamics and parametrized sub-grid processes (‘physics’).
IUnderstanding coupling of dynamics and physics central to
improving weather and climate models e.g. tropical
performance and diurnal cycle over land.
IComplex models should converge to a state of geostrophic
balance at small Rossby number (Ro). Convergence rate Ro
2,
Cullen (2008).
IDoes this idea extend to the physics? For boundary-layer
parametrization, use Ekman balance.
IUnphysical deviation from Ekman balance could be used to
diagnose numerical coupling problems.
ITest this approach for an idealised baroclinic wave.
2 / 13
![Page 8: Understanding physics-dynamics coupling in weather and climate …emps.exeter.ac.uk/media/universityofexeter/emps/climathnet/Robert... · I Understanding coupling of dynamics and](https://reader033.fdocuments.net/reader033/viewer/2022042401/5f0fe40f7e708231d44665dd/html5/thumbnails/8.jpg)
Introduction
IWeather and climate models partitioned into resolveddynamics and parametrized sub-grid processes (‘physics’).
IUnderstanding coupling of dynamics and physics central to
improving weather and climate models e.g. tropical
performance and diurnal cycle over land.
IComplex models should converge to a state of geostrophic
balance at small Rossby number (Ro). Convergence rate Ro
2,
Cullen (2008).
IDoes this idea extend to the physics? For boundary-layer
parametrization, use Ekman balance.
IUnphysical deviation from Ekman balance could be used to
diagnose numerical coupling problems.
ITest this approach for an idealised baroclinic wave.
2 / 13
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Slaving concept
H
h
Boundary layer (function of Ro)
Dynamics (function of Ro)
?
Slaving
Figure: Illustration of the slaving principle between the dynamics and
boundary layer for Ekman-balanced states. Given the dynamics is a
function of Ro, slaving means that the boundary layer is also a function
of Ro.
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Momementum equation
Scaling momentum equation assuming the boundary layer is slaved
to the dynamics. Slaving means the boundary layer shares the
timescales of the dynamics.
Material Derivativez }| {Ro
Dbu
Dbt+
Coriolisz }| {k⇥ b
u +
Pressure gradientz }| {Ro
Fr
2brb� =
Dragz }| {EkBbu,
b are dimensionless values.
Ro =
U
fLFr =
U
NH
Ek =
K
f h2
B = h2 @
@z
✓bKm
@
@z
◆
U horizontal velocity, L length, f Coriolis parameter, NBrunt-Vaisalla frequency, H troposphere depth, K typical
boundary-layer di↵usion, h mean boundary-layer depth.
Full details in Beare and Cullen 20155 / 13
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Simplified momentum equation
For pressure gradient O(1) and geostrophic limit at small Ro
Ro = Fr
2.
For boundary-layer depths small compared to troposphere, pressure
gradient constant across the boundary layer. Coriolis and drag
terms have the same magnitude
Ek = O(1).
Momentum equations simplifies to
Material Derivativez }| {Ro
Dbu
Dbt+
Coriolisz }| {k⇥ b
u +
Pressure gradientz}|{brb� =
Dragz}|{Bbu .
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Ekman-balanced states
Ekman balance In the limit of Ro ! 0
Coriolisz }| {k⇥ b
u
e
+
Pressure gradientz}|{brb� =
Dragz}|{Bbu
e
,
where
bu
e
is the Ekman momentum.
SGT model Additional trajectory (
bu
s
) variable. The SGT
momentum balance is
Balanced material derivativez }| {Ro
DsbueDbt
+
Coriolisz }| {k⇥ b
u
s
+
Pressure gradientz}|{brb� =
Balanced dragz }| {B(2bu
e
� bu
s
),
where
Ds
Dbt=
@
@bt+
bu
s
.r+
bws@
@bz
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Baroclinic wave.
bu � bug at Ro = 0.15, contour interval
0.02.
Primitive equations
!x
-0.5 0 0.5
!z 0.15
0.3
Ekman balance SGT model
!x
-0.5 0 0.5
!z 0.15
0.3
!x
-0.5 0 0.5
!z 0.15
0.3
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bu � bug at Ro = 0.04, contour interval 0.02.
Primitive equations
!x
-0.5 0 0.5
!z 0.15
0.3
Ekman balance SGT model
!x
-0.5 0 0.5
!z 0.15
0.3
!x
-0.5 0 0.5
!z 0.15
0.3
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Convergence with Ro
Rossby no.10
-210
-1
Dev
iati
on f
rom
bal
ance
10-3
10-2
10-1
Ekman (ue)
SGT (us)
Ro
Ro1.7
Figure: Log-log plot of deviation of primitive equation model from either
Ekman balance ( filled circles) or SGT model (crosses) against Ro.
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Boundary-layer numerical instability
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Disruption of convergence with Ro
Rossby no.0.04 0.06 0.08 0.1
Dev
iati
on f
rom
bal
ance
10-3
10-2
10-1
ControlK-updateWood et al.
Ro1.7
Figure: A log-log plot of deviation of the primitive equation model from
the SGT model against Ro for the di↵erent boundary-layer time-stepping
schemes and a larger time step. 12 / 13
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Summary
IConvergence to Ekman balance challenging test for
physics-dynamics coupling in weather and climate models.
IConvergence with Rossby number shown for a boundary layer
coupled to a baroclinic wave. Varies as Ro with respect to
Ekman balance and Ro
1.7with respect to the SGT model.
INot quite as fast convergence as dynamics-only models (Ro
2).
IConvergence disrupted by unstable numerical time-stepping
methods.
IApproach can be used to design idealised testbeds including
physics.
IExtend ideas to tropical convection.
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Summary
IConvergence to Ekman balance challenging test for
physics-dynamics coupling in weather and climate models.
IConvergence with Rossby number shown for a boundary layer
coupled to a baroclinic wave. Varies as Ro with respect to
Ekman balance and Ro
1.7with respect to the SGT model.
INot quite as fast convergence as dynamics-only models (Ro
2).
IConvergence disrupted by unstable numerical time-stepping
methods.
IApproach can be used to design idealised testbeds including
physics.
IExtend ideas to tropical convection.
13 / 13
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Summary
IConvergence to Ekman balance challenging test for
physics-dynamics coupling in weather and climate models.
IConvergence with Rossby number shown for a boundary layer
coupled to a baroclinic wave. Varies as Ro with respect to
Ekman balance and Ro
1.7with respect to the SGT model.
INot quite as fast convergence as dynamics-only models (Ro
2).
IConvergence disrupted by unstable numerical time-stepping
methods.
IApproach can be used to design idealised testbeds including
physics.
IExtend ideas to tropical convection.
13 / 13
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Summary
IConvergence to Ekman balance challenging test for
physics-dynamics coupling in weather and climate models.
IConvergence with Rossby number shown for a boundary layer
coupled to a baroclinic wave. Varies as Ro with respect to
Ekman balance and Ro
1.7with respect to the SGT model.
INot quite as fast convergence as dynamics-only models (Ro
2).
IConvergence disrupted by unstable numerical time-stepping
methods.
IApproach can be used to design idealised testbeds including
physics.
IExtend ideas to tropical convection.
13 / 13
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Summary
IConvergence to Ekman balance challenging test for
physics-dynamics coupling in weather and climate models.
IConvergence with Rossby number shown for a boundary layer
coupled to a baroclinic wave. Varies as Ro with respect to
Ekman balance and Ro
1.7with respect to the SGT model.
INot quite as fast convergence as dynamics-only models (Ro
2).
IConvergence disrupted by unstable numerical time-stepping
methods.
IApproach can be used to design idealised testbeds including
physics.
IExtend ideas to tropical convection.
13 / 13
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Summary
IConvergence to Ekman balance challenging test for
physics-dynamics coupling in weather and climate models.
IConvergence with Rossby number shown for a boundary layer
coupled to a baroclinic wave. Varies as Ro with respect to
Ekman balance and Ro
1.7with respect to the SGT model.
INot quite as fast convergence as dynamics-only models (Ro
2).
IConvergence disrupted by unstable numerical time-stepping
methods.
IApproach can be used to design idealised testbeds including
physics.
IExtend ideas to tropical convection.
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Cullen, M. J. P. (2008). A comparison of numerical solutions to
the Eady frontogenesis problem. Quart. J. Roy. Meteorol. Soc.,134, 2143–2155.
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