Parameterization of Lakes in NWP:
Description of a Lake Model
and Single-Column Tests
Dmitrii Mironov
German Weather Service, Offenbach am Main, Germany
Frank Beyrich and Erdmann Heise (German Weather Service)
Arkady Terzhevik (Northern Water Problems Research Institute, Petrozavodsk, Russia)
Outline
•
•
•
•
•
The Problem
Basic Idea and Some History
The Lake Model
Results from Single-Column Tests
Conclusions
Lake Parameterizations for NWP and
Climate Modeling Systems
(e.g. Ljungemir et al. 1996, Goyette et al. 2000, Tsuang et al. 2001)
• One-layer models, complete mixing down to the bottom
Neglect stratification  large errors in the surface temperature
• Turbulence closure models, multi-layer (finite-difference)
Describe the lake thermocline better  expensive computationally
A compromise between
physical realism and computational economy
is required
A two layer-model with
a parameterized vertical temperature structure
The Concept
• Put forward by Kitaigorodskii and Miropolsky (1970) to
describe the temperature structure of the oceanic seasonal
thermocline. The essence of the concept is that the temperature
profile in the thermocline can be fairly accurately
parameterised through a “universal” function of dimensionless
depth, using the temperature difference across the thermocline,
Δθ=θs(t)-θb(t), and its thickness, Δh, as appropriate scales of
temperature and depth:
 s (t )   ( z, t )
z  h(t )
  ( ),  
.
 (t )
h(t )
Schematic representation of the temperature
profile in the mixed layer and in the thermocline
b
h
h+h
s
Here, θs(t) is the
temperature of the
mixed layer of
depth h(t), and θb(t)
is the temperature
at the lake bottom,
z=h+Δh.
A close analogy to
the mixed-layer concept
• Using the mixed-layer temperature θs(t) and its thickness h(t)
as appropriate scales, the mixed-layer concept states that
 ( z, t )
z
 ML ( ),  
,
 s (t )
h(t )
where the shape function υML is simply a constant equal to one.
Antecedents
Munk, W. H., and E. R. Anderson, 1948: Notes on a theory of the thermocline.
J. Mar. Res., 7, 276-295.
… the upper layers are stirred until an almost homogeneous layer is formed,
bounded beneath by a region of marked temperature gradient, the
thermocline. … If the wind increases in intensity the thermocline moves
downward, but the characteristic shape of the temperature-depth curve
remains essentially unchanged. (Original authors' italic.)
Ertel, H., 1954: Theorie der thermischen Sprungschicht in Seen. Acta
Hydrophys., Bd. 2, 151-171.
Used an analytical solution to the heat transfer equation. Found that the
ratio of the depth from the upper boundary of the lake thermocline
(“thermische Sprungschicht”) to the bend point of the temperature profile
to the depth from the bend point to the bottom of the thermocline is
constant. In other words, the shape of the temperature-depth curve in the
thermocline is independent of time.
Support through Observations
• Observations in the ocean or seas
(Miropolsky et al 1970, Nesterov and Kalatsky 1975, Kharkov 1977,
Reshetova and Chalikov 1977, Efimov and Tsarenko 1980,
Filyushkin and Miropolsky 1981, Mälkki and Tamsalu 1985,
Tamsalu and Myrberg 1998)
• Observations in lakes
(Zilitinkevich 1991, Kirillin 2002)
• Laboratory experiments
(Linden 1975, Voropaev 1977, Wyatt 1978)
Dimensionless temperature profile in the lake thermocline. Curves
show a polynomial approximation (Kirillin 2002).
Dimensionless temperature profile in the lake thermocline. Points show
data from measurements in Trout Bog (depth=7.7 m). Curves show a
polynomial approximation (Kirillin 2002).
Theoretical Background
• In case of the mixed-layer deepening, dh/dt>0,
a travelling-wave type solution to the heat transfer equation has the
form of it shows a good agreement with observational data
(Barenblatt 1978, Turner 1978, Shapiro 1980, Zilitinkevich et al.
1988, Zilitinkevich and Mironov 1989, Mironov 1990, Zilitinkevich
and Mironov 1992, Kirillin 2001a,b).
• No theoretical explanation in case of the mixed-layer stationary state
or retreat, dh/dt0.
The self-similarity at dh/dt0 is purely phenomenological.
Applications
A number of computationally efficient models based on the self-similar
representation of the temperature profile have been developed and successfully
applied to simulate
• the evolution of the mixed layer and seasonal thermocline in the ocean
(Kitaigorodskii and Miropolsky 1970, Miropolsky 1970, Kitaigorodskii
1970, Kamenkovich and Kharkov 1975, Arsenyev and Felzenbaum 1977,
Kharkov 1977, Filyushkin and Miropolsky 1981),
• the atmospheric convectively mixed layer capped by the temperature
inversion (Deardorff 1979, Fedorovich and Mironov 1995, Mironov 1999,
Pénelon et al. 2001), and
• the seasonal cycle of temperature and mixing in medium-depth fresh-water
lakes (Zilitinkevich 1991, Mironov et al. 1991, Golosov et al. 1998).
The Lake Model
Instead of solving partial differential equations (in z, t) for the temperature and
turbulence quantities (e.g. TKE), the problems is reduced to solving ordinary
differential equations for time-dependent parameters that specify the
temperature profile.
•
•
•
The evolution equations for the mean temperature of the water column, the surface
temperature and the bottom temperature (based on the integral heat budgets of the
mixed layer and of the thermocline) and for the mixed-layer depth (based on the
TKE equation integrated over the mixed layer).
Volumetric character of short-wave radiation heating is accounted for.
The shape function (ζ) per se is not required (it is the shape factor Cθ =∫01(ζ)dζ
that enters the model equations).
As the occasion requires, to additional modules are used to describe
•
the snow-ice cover,
•
the interaction of the water column with bottom sediments.
Both modules are built on the same basic principle, i.e. using the assumed shape of the
temperature-depth curve.
The Lake Model
Instead of solving partial differential equations (in z, t) for the temperature and
turbulence quantities (e.g. TKE), the problems is reduced to solving ordinary
differential equations for time-dependent parameters that specify the
temperature profile.
The evolution equations for
• the mean temperature of the water column
obtained by means of integration of the heat transfer equation over the
water column,
• the surface temperature s(t)
obtained by means of integration of the heat transfer equation over the
water column,
• the bottom temperature b(t)
obtained by means of double integration of the heat transfer equation
over the thermocline.
Volumetric character of short-wave radiation heating is accounted for.
The Mixed-Layer Depth
is computed from the TKE equation integrated over the mixed layer.
• Convective deepening of the mixed layer is described by the
entrainment equation. It incorporates the Zilitinkevich (1975) spin-up
correction term that prevents an unduly fast growth of h(t) when the
mixed-layer is shallow.
• During wind mixing, h(t) is computed from a relaxation-type equation,
where the Zilitinkevich and Mironov (1996) multi-limit formulation is
used to compute the equilibrium mixed-layer depth. It accounts for the
effects of the earth’s rotation, surface buoyancy flux and static stability
in the thermocline.
• The equation for h(t) account for the equilibrium depth of a
convectively mixed layer, where convective motions are driven by
surface cooling, whereas the volumetric radiation heating tends to arrest
the mixed layer deepening (Mironov and Karlin 1989).
The shape function () per se is not required.
It is the shape factor
1
C    ( )d
0
that enters the model equations.
As the occasion requires, to additional modules are used to describe
• the snow-ice cover,
• the interaction of the water column with bottom sediments.
Both modules are built on the same basic principle, i.e. using
the assumed shape of the temperature-depth curve.
Schematic representation of the temperature profile
b(t)
s(t)
h(t)
D
H(t)
(a)
L
H(t)
L
(a) The evolving temperature profile is characterised by five time-dependent
parameters, namely, the temperature θs(t) and the depth h(t) of the mixed layer, the
bottom temperature θb(t), the depth H(t) within bottom sediments penetrated by the
thermal wave and the temperature θH(t) at that depth.
S(t)
I(t) s(t) b(t)
Snow
-HI(t)-HS(t)
-HI(t)
Ice
h(t)
Water
H(t)
D
Sediment
H(t)
(b)
L
L
(b) In winter, four more variables are computed, namely, the temperature θS(t) at the
air-snow interface, the temperature θI(t) at the the snow-ice interface, the snow
thickness HS(t) and the ice thickness HI(t).
The results is basically
a phenomenological model
that relies on
“verifiable empiricism’’
but still incorporates much of the
essential physics.
Important! The model does not require (re-)tuning.
The Atmospheric Surface Layer
Parameterization Scheme
• Monin-Obukhov similarity theory to compute turbulent fluxes
of momentum and of sensible and latent heat
• the fetch-dependent Charnock parameter
• the Re0-dependent (aerodynamic roughness Reynolds number)
roughness parameters for scalar quantities
• Nu  Ra1/3 heat-mass transfer law in free convection (low
wind, unstable stratification)
• molecular fluxes in strong static stability
Lake Specific Parameters
• geographical latitude
• the lake depth
• typical wind fetch
• optical characteristics of lake water (extinction
coefficients with respect to short-wave radiation)
• depth of the thermally active layer of bottom sediments,
temperature at that depth (cf. soil model parameters)
Forcing
Given by the driving atmospheric model or known from observations
• the rate of snow accumulation
• short-wave radiation flux
• long-wave radiation flux from the atmosphere
Computed as part of the solution (depend on lake surface temperature)
• long-wave downward radiation flux from the surface
• fluxes of momentum and of sensible and latent heat
• for ice-covered lakes, surface albedo
Kossenblatter See (Germany, 52 N). LITFASS-98.
• depth = 1.2 m
• wind fetch = 1 km
• turbid water, =5 m-1
• both short-wave and long-wave atmospheric radiation fluxes
are measured
• measured water surface temperature (10 min apart)
• measured fluxes of momentum and of sensible and latent (!)
heat (30-min averages)
Kossenblatter See. 3-6 June 1998.
0.4
*
u , m s-1
0.3
0.2
0.1
0
0
12
24
36
48
time, h
Friction velocity in the surface air layer
• Symbols - measured
• Line - computed
60
72
84
96
Kossenblatter See. 3-6 June 1998.
40
•
•
•
•
0
se
Q , W m-2
20
-20
-40
0
12
24
36
48
60
72
84
96
60
72
84
96
time, h
0
-100
-150
la
Q , W m-2
-50
-200
-250
-300
0
12
24
36
48
time, h
Sensible heat flux
Latent heat flux
Symbols – measured
Lines– computed
Kossenblatter See. 3-6 June 1998.
-200
Q
lwa
, W m-2
-250
-300
-350
-400
-450
0
12
24
36
48
time, h
Long-wave atmospheric radiation flux
• Symbols - measured
• Line - computed
60
72
84
96
Kossenblatter See. 8-21 June 1998.
26
25
s-f , K
24
23
22
21
20
19
18
0
48
96
144
192
time, h
Water surface temperature
• Dots - measured
• Line - computed
240
288
336
Kossenblatter See. 8-21 June 1998.
26
•
25
s-f , K
24
•
•
23
22
Water surface temperature
Dots – measured
Line – computed
21
20
19
18
0
48
96
144
192
240
288
336
time, h
26
•
•
Stable stratification
25
-f , K
24
23
•
22
21
20
19
18
0
48
96
144
192
time, h
240
288
336
Surface temperature
Mean temperature of the
water column
Bottom temperature
computed with the lake
model
Kossenblatter See. 8-21 June 1998.
0.5
0.3
*
u , m s-1
0.4
0.2
0.1
0
216
240
264
time, h
Friction velocity in the surface air layer
• Symbols - measured
• Line - computed
288
312
Kossenblatter See. 8-21 June 1998.
•
0.5
•
•
0.3
*
u , m s-1
0.4
Friction velocity in the
surface air layer
Symbols – measured
Line – computed
0.2
0.1
0
216
240
264
288
312
time, h
Stable stratification
0.01
0.005
0
•
•
0.015
*
*
u , w , m s-1
0.02
216
240
264
time, h
288
312
Friction velocity in the
surface layer of watr
Convective velocity scale
computed with the lake
model
Kossenblatter See. 8-21 June 1998.
20
se
Q , W m-2
0
•
•
•
•
-20
-40
-60
-80
216
240
264
288
312
288
312
time, h
0
-100
-150
la
Q , W m-2
-50
-200
-250
-300
216
240
264
time, h
Sensible heat flux
Latent heat flux
Symbols – measured
Lines– computed
Kossenblatter See (Germany, 52 N). 1999.
• depth = 1.2 m
• wind fetch = 1 km
• turbid water, =5 m-1
• both short-wave and long-wave atmospheric radiation fluxes
are measured
• measured water surface temperature (10 min apart)
• measured fluxes of momentum and of sensible heat (sonic, not
corrected, 10-min averages)
Water Surface Temperature
•
•
•
•
Dotted – measured
Solid – computed
(a) 6 August 1999
(b) 13 August 1999
Water Surface Temperature
•
•
•
•
Dotted – measured
Solid – computed
(c) 15 August 1999
(d) 12 October 1999
Fluxes
Kossenblatter See
6 August 1999
• (a) Friction velocity
in the surface air
layer [ms-1]
• (b) ”Virtual” sensible
heat flux [Wm-2]
Fluxes
Kossenblatter See
13 August 1999
• (a) Friction velocity
in the surface air
layer [ms-1]
• (b) ”Virtual” sensible
heat flux [Wm-2]
Fluxes
Kossenblatter See
15 August 1999
• (a) Friction velocity
in the surface air
layer [ms-1]
• (b) ”Virtual” sensible
heat flux [Wm-2]
Fluxes
Kossenblatter See
12 October 1999
• (a) Friction velocity
in the surface air
layer [ms-1]
• (b) ”Virtual” sensible
heat flux [Wm-2]
Lake Swente (Latvia, 56 N). 1961-1964.
• depth = 17.5 m
• wind fetch = 3 km
• transparent water, =0.3 m-1
• computed atmospheric radiation fluxes
• climatologically mean forcing
• measured water temperature at a number of depths
• no flux measurements
Lake Swente. Perpetual year.
20
-f , K
15
10
5
Ice
0
0
60
120
180
240
300
360
time, day
•
•
Surface temperature, mean temperature of the water column, bottom temperature
Symbols – measured, lines – modelled
Lake Swente. Initial state from measurements.
20
-f , K
15
10
5
0
90
120
150
180
210
240
270
300
330
360
time, day
•
•
Surface temperature, mean temperature of the water column, bottom temperature
Symbols – measured, lines – modelled
Ryan Lake (USA, 45 N). 1989-1990.
• depth = 9 m
• wind fetch = 0.5 km
• turbid water, =5 m-1
• measured short-wave atmospheric radiation flux, computed long-wave
atmospheric radiation flux
• some humidity data are missing
• measured water temperature at a number of depths (20 min apart for
the entire period)
• no turbulent flux measurements
Lake Ryan. November 1989 – November 1990.
35
30
-f , K
25
20
15
10
Ice
5
0
-30
0
60
120
180
240
300
time, day
•
•
Surface temperature, mean temperature of the water column, bottom temperature
Dotted – measured, solid – modelled
Lake Ryan, 1989-1990. July 1990.
35
30
-f , K
25
20
15
10
5
0
180
190
200
210
time, day
•
•
Surface temperature, mean temperature of the water column, bottom temperature
Dotted – measured, solid – modelled
Lake Ryan. April – November 1990.
35
30
? (sensor malfunction)
s-f , K
25
20
15
10
5
0
90
120
150
180
210
time, day
• Water surface temperature
• Measured vs. modelled
240
270
300
Lake Ryan. April – November 1990.
20
m-f , K
15
10
5
0
90
120
150
180
210
time, day
• Mean temperature of the water column
• Measured vs. modelled
240
270
300
Lake Ryan. December 1989.
0
i-f , K
-5
-10
-15
-20
-25
-30
-20
-10
time, day
• Solid - modelled ice surface temperature
• Dotted - temperature measured with the uppermost sensor
0
Lake Ryan. December 1989.
0.8
i
h ,m
0.6
0.4
0.2
0
-30
0
30
time, day
• Modelled ice thickness
60
90
Conclusions
• A lake model is developed that offers a good compromise
between physical realism and computational economy
• The model does not require (re-)tuning
• Single-column tests of the lake model (including the surface
air layer parameterization scheme) show promising results
• Future work:
implementation and testing of the new lake model in a 3D
NWP system environment
Acknowledgements: Chris Ellis and Heinz Stefan, St. Anthony Falls
Hydraulic Laboratory, University of Minnesota, USA, data from
measurements in Ryan Lake.
EU Commissions, Project INTAS-01-2132.