Parameterization of Lakes in NWP: Description of a Lake Model and Single-Column Tests

Parameterization of Lakes in NWP: Description of a Lake Model

and Single-Column Tests

Dmitrii MironovGerman 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 economyis 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:

.)(

)(),(

)(

),()(

th

thz

t

tzts

Schematic representation of the temperature profile in the mixed layer and in the thermocline

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.

h

h+h

s

b

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

,)(

),()(

),(

th

z

t

tzML

s

where the shape function υML is simply a constant equal to one.

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.

Antecedents

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/dt0.

The self-similarity at dh/dt0 is purely phenomenological.

Applications

• 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).

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 Lake Model

• 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).

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.

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

• the mean temperature of the water columnobtained 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.

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

Volumetric character of short-wave radiation heating is accounted for.

The Mixed-Layer Depth

• 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).

is computed from the TKE equation integrated over the mixed layer.

The shape function () per se is not required.

• the snow-ice cover,

• the interaction of the water column with bottom sediments.

It is the shape factor

Both modules are built on the same basic principle, i.e. using the assumed shape of the temperature-depth curve.

As the occasion requires, to additional modules are used to describe

that enters the model equations.

dC )(1

0

Schematic representation of the temperature profile

(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)

b(t)

(a)

L

H

(t)

h(t)

D

L

H(t)

(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).

s(t)

b(t)

I(t)

S(t)

(b)

L

H

(t)

h(t)

D

L

H(t)

-HI(t)

-HI(t)-H

S(t)

Snow

Ice

Water

Sediment

The results is basically

a phenomenological model

Important! The model does not require (re-)tuning.

that relies on

“verifiable empiricism’’

but still incorporates much of the

essential physics.

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

• the rate of snow accumulation

• short-wave radiation flux

• long-wave radiation flux from the atmosphere

Given by the driving atmospheric model or known from observations

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.

Friction velocity in the surface air layer

• Symbols - measured

• Line - computed

0 12 24 36 48 60 72 84 960

0.1

0.2

0.3

0.4

time, h

u*, m

s-1


• Sensible heat flux

• Latent heat flux

• Symbols – measured

• Lines– computed

0 12 24 36 48 60 72 84 96

-40

-20

0

20

40

time, h

Qse

, W m

-2

0 12 24 36 48 60 72 84 96-300

-250

-200

-150

-100

-50

0

time, h

Qla

, W m

-2


Long-wave atmospheric radiation flux


• Line - computed

0 12 24 36 48 60 72 84 96-450

-400

-350

-300

-250

-200

time, h

Qlw

a, W m

-2


Water surface temperature

• Dots - measured

• Line - computed

0 48 96 144 192 240 288 33618

19

20

21

22

23

24

25

26

time, h

s-

f , K


• Water surface tempe-rature

• Dots – measured

• Line – computed

• Surface temperature

• Mean temperature of the water column

• Bottom temperature

computed with the lake model

0 48 96 144 192 240 288 33618

19

20

21

22

23

24

25

26

time, h

s-

f , K

0 48 96 144 192 240 288 33618

19

20

21

22

23

24

25

26

time, h

-

f , K

Stable stratification


Friction velocity in the surface air layer


• Line - computed

216 240 264 288 3120

0.1

0.2

0.3

0.4

0.5

time, h

u* ,

m s

-1


• Friction velocity in the surface air layer


• Line – computed

• Friction velocity in the surface layer of watr

• Convective velocity scale

computed with the lake model

Stable stratification

216 240 264 288 3120

0.1

0.2

0.3

0.4

0.5

time, h

u* ,

m s

-1

216 240 264 288 3120

0.005

0.01

0.015

0.02

time, h

u* ,

w* ,

m s

-1


• Sensible heat flux

• Latent heat flux


• Lines– computed

216 240 264 288 312-80

-60

-40

-20

0

20

time, h

Qse

, W

m-2

216 240 264 288 312-300

-250

-200

-150

-100

-50

0

time, h

Qla

, W

m-2

Kossenblatter See (Germany, 52 N). 1999.

• depth = 1.2 m



• 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



Fluxes

Kossenblatter See

15 August 1999



Fluxes

Kossenblatter See

12 October 1999



Lake Swente (Latvia, 56 N). 1961-1964.

• depth = 17.5 m


• 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.

• Surface temperature, mean temperature of the water column, bottom temperature

• Symbols – measured, lines – modelled

0 60 120 180 240 300 360

0

5

10

15

20

time, day

-

f , K

Ice

Lake Swente. Initial state from measurements.


• Symbols – measured, lines – modelled

90 120 150 180 210 240 270 300 330 360

0

5

10

15

20

time, day

-

f , K

Ryan Lake (USA, 45 N). 1989-1990.

• depth = 9 m

• wind fetch = 0.5 km


• 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.


• Dotted – measured, solid – modelled

-30 0 60 120 180 240 3000

5

10

15

20

25

30

35

time, day

-

f , K

Ice

Lake Ryan, 1989-1990. July 1990.


• Dotted – measured, solid – modelled

180 190 200 2100

5

10

15

20

25

30

35

time, day

-

f , K

Lake Ryan. April – November 1990.

• Water surface temperature

• Measured vs. modelled

90 120 150 180 210 240 270 3000

5

10

15

20

25

30

35

time, day

s-

f , K

? (sensor malfunction)

Lake Ryan. April – November 1990.

• Mean temperature of the water column

• Measured vs. modelled

90 120 150 180 210 240 270 3000

5

10

15

20

time, day

m

-f ,

K

Lake Ryan. December 1989.

• Solid - modelled ice surface temperature • Dotted - temperature measured with the uppermost sensor

-30 -20 -10 0

-25

-20

-15

-10

-5

0

time, day

i-

f , K

Lake Ryan. December 1989.

• Modelled ice thickness

-30 0 30 60 900

0.2

0.4

0.6

0.8

time, day

hi ,

m

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.

Parameterization of Lakes in NWP: Description of a Lake Model and Single-Column Tests

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