The mixed-layer model: a simplified model for the convective boundary layer

Chiel van Heerwaarden Max Planck Institute for Meteorology

Max-Planck-Institut für Meteorologie

The goal of this lecture

• Put some of the basic concepts explained in previous lectures into practice

• Derive and use a simple model for the daytime convective boundary layer

• Application of the model in different topics in our practical on Friday:

• Basic land-atmosphere interactions (Chiel)

• The role of CO2 in the coupled land-atmosphere system (Jordi Vilà)

• The link between boundary-layer clouds and the surface energy balance (Bart van Stratum)

• Other lectures

• The urban heat island (Gert-Jan Steeneveld)

• …

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Measured profiles of potential temperature

• AMMA Campaign (Niamey, Niger, West-Africa), 22 June 2006

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q (K)

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8h35 UTC11h33 UTC14h51 UTC17h40 UTC

Simulation of a growing convective boundary layer

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Potential temperature in the convective boundary layer

• Averaged temperature profiles are acquired from the simulations

• We can validate a derived model against simulation data

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q (K)

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z(m

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Potential temperature in the convective boundary layer

• We simplify the CBL and need three equations

• Two parameters go into this equation: the surface flux and the lapse rate

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q (K)

0

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z(m

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⌦✓↵

�✓h

�✓

Potential temperature turbulent flux in the CBL

• The flux profile is assumed to be linear

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�0.02 0.00 0.02 0.04 0.06 0.08 0.10w0

q

0 (K m s�1)

0

200

400

600

800

1000

1200

1400

z(m

)

first hoursecond hourthird hour

Potential temperature turbulent flux in the CBL

• The flux profile is assumed to be linear

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�0.02 0.00 0.02 0.04 0.06 0.08 0.10w0

q

0 (K m s�1)

0

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z(m

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w0✓00

w0✓0h

The starting point of for the derivation of the model

• Conserved variables: we are going to use potential temperature

• A parcel of air that is being advected upwards or downwards does not change temperature as an effect of pressure change

• Reynolds averaging: the turbulent flux of potential temperature

• Even when there is no mean vertical wind, heat is transported vertically

• The conservation equation of potential temperature in the dry atmospheric boundary layer can be written as:

• Dennis’ talk: we need to know the fluxes

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w0✓0

@✓

@t= �@w0✓0

@z

Creating a budget equation for a layer in the atmosphere

• How to derive the temperature budget of a vertical layer in the atmosphere?

• We replace the first term on the right hand side with the conservation equation

!

!

• With < brackets > we define the bulk potential temperature

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@✓

@t= �@w0✓0

@z

d

dt

Z b

a✓ dz =

Z b

a

@✓

@tdz + ✓b

db

dt� ✓a

da

dt

d

dt

Z b

a✓ dz = �w0✓0b + w0✓0a + ✓b

db

dt� ✓a

da

dt

Z b

a✓ dz ⌘

⌦✓↵(b� a)

Apply the equation from surface to boundary layer top

• We assume that we take the budget from surface to the CBL top

• We have a discontinuous value, so we have to work with a limit.

!

!

!

• If we take into account the assumed profiles, we can cancel two terms, and take the limit of epsilon to zero:

• This is the same as a budget equation for a box of fixed size

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d

dt

�⌦✓↵(h� ✏)

�= �w0✓0h�✏ + w0✓00 + ✓h�✏

d (h� ✏)

dt

⌦✓↵ d (h� ✏)

dt+ (h� ✏)

d⌦✓↵

dt= �w0✓0h�✏ + w0✓00 + ✓h�✏

d (h� ✏)

dt

d⌦✓↵

dt=

w0✓00 � w0✓0hh

The budget of the top of the convective boundary layer

• We take the budget over the CBL top to get the evolution equation for the CBL depth h

!

!

• If we take the again the limit of epsilon going to zero, we get

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0 = w0✓0h +�✓hdh

dt

dh

dt= �w0✓0h

�✓h

d

dt

Z h+✏

h�✏✓ dz = �w0✓0h+✏ + w0✓0h�✏ + ✓h+✏

d (h+ ✏)

dt� ✓h�✏

d (h� ✏)

dt

The equation for the evolution of the jump is missing

• This equation is a combination of the previous two

!

!

• Now we have three equations, but still one problem to solve…

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d�✓

dt= �✓

dh

dt�

d⌦✓↵

dt

d⌦✓↵

dt=

w0✓00 � w0✓0hh

dh

dt= �w0✓0h

�✓h

d�✓

dt= �✓

dh

dt�

d⌦✓↵

dt

The entrainment closure

• The entrainment of potential temperature is independent of the size of the jump and the stratification

• Analysis of the kinetic energy budget suggests a simple parameterization:

!!

• The value of the constant varies between 0.12 - 0.2

• We are have derived the model!

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w0✓0h = ��w0✓00

d⌦✓↵

dt=

(1 + �)w0✓00h

dh

dt=

�w0✓00�✓h

d�✓

dt= �✓

dh

dt�

d⌦✓↵

dt

Next session: three options

• Choose one of the three following options for Friday’s practical

• The coupled land-atmosphere system (Chiel):

• Students will go into detail in explaining the feedbacks in the coupled system. Emphasis will be given to the assessment of the relative importance of different feedbacks and this dependence to different environmental conditions.

• Soil moisture and boundary-layer clouds (Bart):

• Soil moisture influences the partitioning of the surface fluxes, promoting boundary layer cloud development through both moistening of the atmospheric boundary layer, and boundary-layer growth. Students will study the (non-linear) relation between soil moisture and the boundary layer.

• Dynamic vegetation and carbon dynamics (Jordi):

• During the practical we will analyze and discuss the sensitivity of the vegetation-atmosphere system to CO2-fertilization, warming, dimming and wind stilling.

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