Thoughts on the Parameterization of Physical Processes
(by way of an example)
Murray D. MacKay
Environment Canada
Example: Modelling the Temperature Profile of a Lake
solar energy
light attenuation
Temperature
Heat Equation
∂ (CwT )
∂F ∂Q
=−
−
∂t
∂z ∂z
light extinction follows Beer’s Law
Example: Modelling the Temperature Profile of a Lake
solar energy
light attenuation
Heat Equation
∂ (CwT )
∂F ∂Q
=−
−
∂t
∂z ∂z
Temperature
Fundamental Law of Heat Conduction
∂T
F ( z) = − K m
∂z
Example: Modelling the Temperature Profile of a Lake
solar energy
Temperature profiles: Wellington Reservoir
Heat Equation in a Fluid
∂ (CwT )
∂F ∂Q Cw∂ (w′T ′)
=−
−
−
∂t
∂z ∂z
∂z
turbulent fluctuations
Problem: We need to parameterize turbulence.
A
Approach
h1
1: Fi
Firstt O
Order
d T
Turbulence
b l
Cl
Closure
• assumes turbulent eddy motion transports heat
g
y to molecular diffusion
analogously
• requires eddy size to be small
• does not parameterize turbulence itself – just the
impact of turbulence on temperature profile
Heat Equation in a Fluid
∂ (CwT )
∂F ∂Q Cw∂ (w′T ′)
=−
−
−
∂z ∂z
∂t
∂z
∂ ⎛
∂T ⎞
⎜ Km
⎟
∂z ⎝
∂z ⎠
Property of fluid
(basically invariant)
∂ ⎛ ∂T ⎞
⎜ Ke
⎟
∂z ⎝
∂z ⎠
Property of flow
Ke = constant
Ekman Theory
Ke = Ke(z)
constant shape
Ke = Ke(z,t)
most general
Henderson – Sellers (1985)
(
)[
kˆu* z
*
2
K e (z, t ) =
exp − k z 1 + 37 Ri
P0
drift decay coefficient
k * = 6.6 sin ϑU −1.84
[
]
−1
(
(
2
− 1 + 1 + 40 N kˆ 2 z 2 / u* exp − 2k * z
Ri =
20
2
))]
1
2
Problem: We need to parameterize turbulence.
Approach 2: Integrated Turbulent Kinetic Energy (TKE)
• assumes uniform well mixed layer at surface
• does parameterize turbulence itself as well as the
impact of turbulence on temperature profile
T
Temperature
profiles:
fil
W
Wellington
lli
R
Reservoir
i
z=0
surface
h dE s
2 dt
mixed layer
z=h
E s dh
2 dt
thermocline
metalimnion
h dE s
= Fq − Fd − Fi
2 dt
Es dh
= Fi + Fs − Fp − FL
2 dt
Governing Equations
wind and buoyancy
y
y forcing
g
z=0
1 3
Fq = (w* + cn3u*3 )
2
surface
h dE s
2 dt
mixed layer
z=h
E s dh
2 dt
thermocline
metalimnion
h dE s
= Fq − Fd − Fi
2 dt
Es dh
= Fi + Fs − Fp − FL
2 dt
Governing Equations
z=0
Fq
h dE s
2 dt
mixed layer
z=h
E s dh
2 dt
surface
dissipation
p
1
Fd = ce E s3 / 2
2
thermocline
metalimnion
h dE s
= Fq − Fd − Fi
2 dt
Es dh
= Fi + Fs − Fp − FL
2 dt
Governing Equations
z=0
Fq
surface
h dE s
2 dt
mixed layer
z=h
Fd
turbulent transport
1
Fi = c f E s3 / 2
2
E s dh
2 dt
thermocline
metalimnion
h dE s
= Fq − Fd − Fi
2 dt
Es dh
= Fi + Fs − Fp − FL
2 dt
Governing Equations
Fq
z=0
h dE s
2 dt
mixed layer
surface
Fd
Fi
shear production
z=h
E s dh
2 dt
metalimnion
Fs =
thermocline
1
dh
cs Δu 2
2
dt
h dE s
= Fq − Fd − Fi
2 dt
Es dh
= Fi + Fs − Fp − FL
2 dt
Governing Equations
Fq
z=0
h dE s
2 dt
mixed layer
surface
Fd
Fi
cold water entrainment
z=h
metalimnion
Fs
E s dh
2 dt
h dE s
= Fq − Fd − Fi
2 dt
Es dh
= Fi + Fs − Fp − FL
2 dt
thermocline
1
dh
Fp = − αghΔT
2
dt
Governing Equations
Fq
z=0
surface
h dE s
2 dt
mixed layer
Fd
Fi
leakage
z=h
metalimnion
Fs
E s dh
2 dt
thermocline
Fp
1
FL = cL Es3 / 2
2
h dE s
= Fq − Fd − Fi
2 dt
Es dh
= Fi + Fs − Fp − FL
2 dt
Governing Equations
z=0
T0
wind and buoyancy forcing
d
TKE
dt
mixed layer
dissipation
Tmix
turbulent transport
z h
z=h
metalimnion
•
shear production
d
h
dt
cold water entrainment
leakage
well know sources/sinks of turbulent kinetic energy
govern depth of mixed layer
Tsfc (oC
C)
L239 Surface Temperature
July 8
Julian Dayy
Oct 17
wind – driven deepening
z (m)
restratification
wind – driven deepening
Tsfc
TKE Forcing
1 3
Fq = (w* + cn3u*3 )
2
wind stress
TKE
wind and buoyancy forcing
z=0
TKE forcing
d
TKE
dt
dissipation
mixed layer
turbulent transport
Thermocline forcing
d
h
d
dt
z=h
cold water entrainment
metalimnion
shear production
leakage
wind – driven deepening
Tsfc
Initial
t a Conditions
Co d t o s
0
20:00 July 8
1
z (m
m)
2
3
4
TKE
5
6
TKE forcing
7
12
14
16
18
T (oC)
Thermocline forcing
20
22
24
wind – driven deepening
Tsfc
0
08:00 July 9
1
08:00 July 9
z (m
m)
2
3
4
TKE
5
6
TKE forcing
7
12
14
16
18
T (oC)
Thermocline forcing
20
22
24
wind – driven deepening
Tsfc
13:30 July 10
0
13:30 July 10
1
z (m
m)
2
TKE
3
4
5
TKE forcing
6
7
12
14
16
18
T
Thermocline forcing
(oC)
20
22
24
CLASS Lake Module
z=0
T0
wind and buoyancy forcing
d
TKE
dt
mixed layer
dissipation
Tmix
turbulent transport
z h
z=h
metalimnion
shear production
d
h
dt
cold water entrainment
leakage
Henderson – Sellers (1985)
)[
kˆu* z
*
2
K=
exp − k z 1 + 37 Ri
P0
(
drift decay coefficient
k * = 6.6 sin ϑU −1.84
[
]
−1
(
(
2
− 1 + 1 + 40 N kˆ 2 z 2 / u* exp − 2k * z
Ri =
20
2
))]
1
2
z (m)
restratification
restratification
TKE
TKE Forcing
Fq =
TKE forcing
1 3
(
w* + cn3u*3 )
2
buoyancy
[
]
2 h
⎡
⎤
− L* − H S − H E − ⎢Q * + Q ( h) − ∫ Qdz ⎥
h 0
⎣
⎦
restratification
0
08:50 July 15
1
TKE
z (m)
2
3
4
5
TKE forcing
6
7
12
14
16
18
T (oC)
20
22
24
restratification
0
20:00 July 15
1
TKE
z (m)
2
3
4
5
TKE forcing
6
7
12
14
16
18
T (oC)
20
22
24
restratification
0
08:00 July 16
1
TKE
z (m)
2
3
4
5
TKE forcing
6
7
12
14
16
18
T (oC)
20
22
24
restratification
0
20:00 July 16
1
TKE
z (m)
2
3
4
5
TKE forcing
6
7
12
14
16
18
T (oC)
20
22
24
restratification
0
08:00 July 17
1
TKE
z (m)
2
3
4
5
TKE forcing
6
7
12
14
16
18
T (oC)
20
22
24
restratification
0
20:00 July 17
1
TKE
z (m)
2
3
4
5
TKE forcing
6
7
12
14
16
18
T (oC)
20
22
24
restratification
0
20:00 July 18
1
TKE
z (m)
2
3
4
5
TKE forcing
6
7
12
14
16
18
T (oC)
20
22
24
restratification
0
08:00 July 19
1
TKE
z (m)
2
3
4
5
TKE forcing
6
7
12
14
16
18
T (oC)
20
22
24
restratification
0
20:00 July 19
1
TKE
z (m)
2
3
4
5
TKE forcing
6
7
12
14
16
18
T (oC)
20
22
24
Co c us o s
Conclusions
• Where possible
possible, begin from fundamental law (e.g.
(e g conservation
of energy, mass, etc.).
• Clearly state (and understand) the assumptions you need to simplify
the problem (e.g. turbulent eddies small for eddy diffusivity concept).
• It’s better to parameterize processes that cause an effect than it is to
parameterize the effect itself (e.g. rather than parameterize a temperature
profile, it’s better to parameterize processes that move heat)
LAKE1D – Turbulence Model
z=0
h dE s
2 dt
mixed layer
z=h
metalimnion
E s dh
2 dt
thermocline
d ⎛1
⎞ h dE s E s dh
+
⎜ hE s ⎟ =
2 dt
dt ⎝ 2
⎠ 2 dt
= Fq − Fd + Fs − Fp