Boundary layer dynamics in a simple model for convectively coupled gravity waves

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Boundary layer dynamics in a simple model for

convectively coupled gravity waves

Michael L. Waite

Boualem Khouider

University of Victoria

With thanks to: Andy Majda, Norm McFarlane, Adam Monahan, Bjorn Stevens

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Convectively coupled Kelvin waves

◮

Eastward propagation

◮

Wavelength: _{λ ∼} O(1000) km

◮

Phase speed: c _∼ 10-20 m/s

(CLAUS brightness temperature from Kiladis et al. 2009)

(3)

Vertical structure and downdrafts

(Khouider & Majda 2008)

(4)

Outline

1. Background

Khouider–Majda multicloud model

2. Multicloud model with boundary layer dynamics

Boundary layer equations

Coupling to free troposphere

Free troposphere equations

3. Waves and stability

Basic state

Moist basic state instabilities

Dry basic state instabilities

4. Conclusions

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1. Background

Outline

1. Background

Khouider–Majda multicloud model

2. Multicloud model with boundary layer dynamics

Boundary layer equations

Coupling to free troposphere

Free troposphere equations

3. Waves and stability

Basic state

Moist basic state instabilities

Dry basic state instabilities

4. Conclusions

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1. Background Khouider–Majda multicloud model

Idealized models for convectively coupled waves

◮

_n ₌ 1: Deep: e.g. Emanuel (1987), Yano et al. (1995)

◮

_n ₌ _{2: Deep} ₊ stratiform: Mapes (2000), Majda & Shefter (2001)

◮

_n ₌ _{2: Deep} ₊ _stratiform ₊ congestus: Khouider & Majda (2006, 2008)

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Khouider–Majda multicloud model structure

q

2

1 2

z

0

−h H

θ

u

θ

u

θ

_eb

1

◮

Free troposphere: 0 _≤ z _≤ H

◮

Boundary layer: ₋ h _≤ z < 0

◮

_H ₌ _{16 km}

◮

_h ₌ _{500 m}

q = ¹

H

Z

H 0

(q

v

₋ q

_v

) dz

(·)

^b

⁼ ¹ _h

Z

0

−^h

(·) ^dz

∂u

j

∂t ^{= ∇θ}

^j

^{+ . . .}

∂θ

1

∂t ^{= ∇ ·} ^u

¹

⁺ ^H

^d

^{+ ξ}

^s

^H

^s

^{+ ξ}

^c

^H

^c

^{+ . . .}

∂θ

2

∂t ⁼

1

4 ∇ · _u

₂

₋ _H

_s

₊ _H

_c

_{+ . . .}

(8)

Boundary layer

◮

Boundary layer in Khouider & Majda (2006, 2008) is purely thermodynamical:

∂θ

eb

∂t ⁼ ^S ⁻ ^D,

◮

S is moistening by surface fluxes

◮

D is drying by convective downdrafts: D _∝ m

₀

(1 + µ(H

s

₋ H

c

))

⁺

◮

No environmental downdrafts because w = 0 at the top of the boundary layer.

◮

Environmental downdrafts require boundary layer dynamics.

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2. Multicloud model with boundary layer dynamics

Outline

1. Background

Khouider–Majda multicloud model

2. Multicloud model with boundary layer dynamics

Boundary layer equations

Coupling to free troposphere

Free troposphere equations

3. Waves and stability

Basic state

Moist basic state instabilities

Dry basic state instabilities

4. Conclusions

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2. Multicloud model with boundary layer dynamics Boundary layer equations

Boundary layer equations

◮

Derive bulk equations following Stevens (TCFD 2006).

◮

Average Reynolds equations over ₋ h _≤ z _≤ 0 to get equations for u

_b

, θ

b

^, θ

eb

^.

◮

Assume constant h

◮

Write turbulent fluxes as:

◮

_h _w

′

_u

′

i

s

= − ^{c U u}

b

◮

_h _w

′

_φ

′

i

s

= ^h

τ

e

(φ

s

− φ

b

)

◮

_hw

′

_u

′

_i

t

= ^h

τ

T

(u

b

− u

t

)

◮

_h _w

′

_φ

′

i

t

= M

u

(φ

b

− φ

t

) − ^M

d

(φ

m

− φ

t

)

◮

Subscripts denote reference levels:

◮

s: surface z = −h

◮

t: top of boundary layer z _→ 0

⁺

◮

m: mid-troposphere

(Following Raymond 1995)

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2. Multicloud model with boundary layer dynamics Boundary layer equations

Boundary layer equations

∂u

b

∂t ⁺ ^u

^b

^{· ∇} ^u

^b

^{= −∇} ^p

^b

⁻ ^c

U

h ^u

^b

⁻

1 h

„ h

τ

T

+ h ^{∇ ·} u

b

«

∆

t

u

∂θ

b

∂t ⁺ ^u

^b

^{· ∇} ^θ

^b

^{= −} ^Q

^Rb

⁺

∆

s

θ

τ

e

⁻

M

d

h ^∆

^m

^θ ⁻

1 h ^(M

^u

⁻ ^M

^d

⁺ ^h ^{∇ ·} ^u

^b

^{) ∆}

^t

^θ

∂θ

eb

∂t ⁺ ^u

^b

^{· ∇} ^θ

^eb

^{= −} ^Q

^Rb

⁺

∆

s

θ

e

τ

e

⁻

M

d

h ^∆

^m

^θ

^e

⁻

1 h ^(M

^u

⁻ ^M

^d

⁺ ^h

∇ · _u

_b

_{) ∆}

_t

_θ

_e

◮

_surface, downdrafts, entrainment, fluxes (e.g. Stevens 2006)

◮

Downward differences: ∆

s

_{φ ≡ φ}

s

_{− φ}

b

, ∆

t

φ ≡ φ

b

− φ

t

, ∆

m

_{φ ≡ φ}

b

− φ

^m

◮

Redefine entrainment velocities:

E

u

_≡

„ _h

τ

T

+ h ^{∇ ·} u

b

«

+

E _{≡ (} M

u

₋ M

d

+ h ^{∇ ·} u

b

)

⁺

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2. Multicloud model with boundary layer dynamics Coupling to free troposphere

Continuity of w

◮

Discontinuity of w at interface between boundary layer and free troposphere:

◮

Introduce barotropic u

0

in troposphere with ^{∇ ·} u

0

≡ −δ ∇ · ^u

b

, where _{δ ≡} h/H

z

0

−h H

θ

u

θ

u

θ_eb ^ubθb

u0 w0

1 ₂

1 2

q

+

0 500 1000

−0.5 0 16

x [km]

z [km]

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Pressure

◮

Need expressions for p

_b

and p

₀

.

◮

Assume BL pressure is

◮

linear in z: p

b

= ¹

2 ^(p

^s

⁺ ^p

^t

^{) =}

1

2 “

p

s

+ p

0

− ^√ ^2(θ

1

− θ

2

) ^”

◮

and hydrostatic: p

s

= p

0

− ^√ ^2(θ

1

+ θ

2

) − πδθ

b

◮

Divergence of u

b

and u

0

equations yield diagnostic equation for p

b

:

−( ¹ + δ) ∇

²

^p

^b

= ∇

²

^“ ^√ ² ^(θ

¹

^{+ θ}

²

^{) +} ^π ₂ ^{δ θ}

^b

^” ⁺ ^c ^U _H ^{∇ ·} ^u

^b

⁺ ^NL,

◮

For linear waves, get

p

_b

_{≈ −} ^√ 2(θ

1

+ θ

2

) − ^π ₂ ^{δ θ}

b

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Downdrafts

◮

Need expressions for M

u

and M

d

.

◮

Parameterize M

d

with convective and environmental contributions:

M

d

≡ ( ^D

c

+ h ^{∇ ·} u

b

)

⁺

, where D

c

≡ ^m

0

(1 + µ(H

s

− ^H

c

))

⁺

◮

Note: neglecting ^{∇ ·} u

b

in M

d

leads to catastrophic short-wave instability! Recall

E _{≡ (} M

u

− ^M

d

+ h ^{∇ ·} u

b

)

⁺

◮

_{Assume M}

u

∝ ^D

c

(Raymond 1995). Define α

m

≡ ^D

c

/M

u

. Use α

m

= 0.2.

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2. Multicloud model with boundary layer dynamics Free troposphere equations

Free troposphere equations

∂u

0

∂t ⁺ ^u

⁰

^{· ∇} ^u

⁰

^{= −∇} ^p

⁰

^{− ∇ ·} ^u

⁰

^(u

⁰

⁻ ^u

^b

^{) +}

E

u

H ^∆

^t

^u

∂u

j

∂t ⁺ ^u

⁰

^{· ∇} ^u

^j

⁺ ^u

^j

^{· ∇} ^u

⁰

^{= ∇θ}

^j

⁺

√ 2δ

τ

T

∆

t

u

∂θ

1

∂t ⁺ ^u

⁰

^{· ∇} ^θ

¹

⁺

√ 2 ^{∇ ·} u

0

− ∇ · ^u

1

= H

d

+ ξ

s

H

s

+ ξ

c

H

c

₋ Q

R1

∂θ

2

∂t ⁺ ^u

⁰

^{· ∇} ^θ

²

⁺

√ 2

4 ^{∇ ·} ^u

⁰

⁻

1 4 ^{∇ ·} ^u

²

^{= −} ^H

^s

⁺ ^H

^c

⁻ ^Q

^R2

∂q

∂t ⁺ ^u

⁰

^{· ∇} ^q ^{+ ∇ ·}

“ (u

1

+ ˜ λu

2

) ˜ Q ₋ u

0

Q ^˜

0

”

= − ^P ⁺ ^E _H ^∆

^t

^θ

^e

⁺ _H ¹ ^(M

^d

− ^h ^{∇ ·} ^u

b

) ∆

m

θ

e

◮

Note: we’ve neglected:

◮

Baroclinic–baroclinic advection

◮

Barotropic vertical advection of baroclinic modes

◮

_{Heating of} _θ

1

and θ

2

by boundary layer (but included in q equation for conservation)

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3. Waves and stability

Outline

1. Background

Khouider–Majda multicloud model

2. Multicloud model with boundary layer dynamics

Boundary layer equations

Coupling to free troposphere

Free troposphere equations

3. Waves and stability

Basic state

Moist basic state instabilities

Dry basic state instabilities

4. Conclusions

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3. Waves and stability Basic state

Basic state

◮

_{Let f} ₌ _{0, v} ₌ _0, _∂

_y

₌ ₀

◮

Linearize around basic state in radiative–convective equilibrium (RCE).

◮

RCE is determined by specifying Q

_R1

and profiles of θ

e

, θ. Solve for heating/cooling and

some parameters.

◮

Regime characterized by θ

eb

− θ

^em

◮

Moist troposphere: θ

eb

− θ

em

= 11 K

◮

Dry troposphere: θ

eb

− θ

em

= 19 K

◮

RCE fixes values for parameters Q

Rb

and τ

e

. Typical values:

◮

_Q

Rb

≈ 5 K/day

◮

_τ

_e

_≈ _{7 h}

◮

_M

d

≈ ^{0.5 cm/s}

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3. Waves and stability Moist basic state instabilities

Moist basic state: moist gravity wave stability

With boundary layer Without boundary layer

0 10 20 30 40 50 60 70 80 90 100

−0.2

−0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

k [ 2π/40000 km⁻¹ ]

Growth rate [ 1/day ]

0 10 20 30 40 50 60 70 80 90 100

−0.2

−0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

k [ 2π/40000 km⁻¹ ]

0 10 20 30 40 50 60 70 80 90 100

−30

−20

−10 0 10 20 30

k [ 2π/40000 km⁻¹ ]

Phase speed [ m/s ]

0 10 20 30 40 50 60 70 80 90 100

−30

−20

−10 0 10 20 30

k [ 2π/40000 km⁻¹ ]

Phase speed [ m/s ]

(19)

Moist basic state: moist gravity wave structure

Most unstable wave: k = 14, λ = 1818 km, c = +15.0 m/s

Depth [ km ]

0 4 8 12 16

0 681 1363 2045 2727

−1 0 1

x [ km ]

qθ_eb θ_b

−1 0

1 M

d^{: Conv} Md^{: Env}

Contours: heating Colours:θ Arrows:(u,w)

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Moist basic state: moist gravity wave budget

θ

eb

budget:

−1 0

1 DD

E Sfc

u

_b

budget:

−1

−0.5 0 0.5 1

PG E Drg

0 681 1363 2045 2727

−1 0 1

x [ km ]

PG: theta1 PG: theta2 PG: thetab

(21)

Moist basic state: fast moist gravity wave stability

With boundary layer Without boundary layer

0 50 100 150 200 250 300 350 400

−0.2

−0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

k [ 2π/40000 km⁻¹ ]

0 50 100 150 200 250 300 350 400

−0.2

−0.1 0 0.1 0.2 0.3 0.4 0.5

k [ 2π/40000 km⁻¹ ]

0 50 100 150 200 250 300 350 400

−30

−20

−10 0 10 20 30

k [ 2π/40000 km⁻¹ ]

Phase speed [ m/s ]

0 50 100 150 200 250 300 350 400

−30

−20

−10 0 10 20 30

k [ 2π/40000 km⁻¹ ]

Phase speed [ m/s ]

(22)

Moist basic state: fast moist gravity wave structure

k = 400, λ = 100 km, c = +26.2 m/s

Depth [ km ]

0 4 8 12 16

0 37 75 112 150

−1 0 1

x [ km ]

qθ_eb θ_b

−1 0 1

M_d: Conv M_d: Env

(23)

Moist basic state: fast moist gravity wave budget

θ

eb

budget:

−1 0

1 DD

E Sfc

u

_b

budget:

−1

−0.5 0 0.5 1

PG E Drg

0 37 75 112 150

−1 0 1

x [ km ]

(24)

Moist basic state: environmental downdrafts

With ^{∇ ·} u

_b

in M

_d

Without ^{∇ ·} u

_b

in M

_d

0 50 100 150 200 250 300 350 400

−0.2

−0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

k [ 2π/40000 km⁻¹ ]

0 50 100 150 200 250 300 350 400

0 1 2 3 4 5

k [ 2π/40000 km⁻¹ ]

0 50 100 150 200 250 300 350 400

−30

−20

−10 0 10 20 30

k [ 2π/40000 km⁻¹ ]

Phase speed [ m/s ]

0 50 100 150 200 250 300 350 400

−30

−20

−10 0 10 20 30

k [ 2π/40000 km⁻¹ ]

Phase speed [ m/s ]

(25)

3. Waves and stability Dry basic state instabilities

Dry basic state: congestus mode stability

With boundary layer Without boundary layer

0 10 20 30 40 50 60 70 80 90 100

0 0.5 1 1.5 2 2.5 3 3.5 4

k [ 2π/40000 km⁻¹ ]

0 10 20 30 40 50 60 70 80 90 100

0 0.5 1 1.5 2 2.5 3 3.5 4

k [ 2π/40000 km⁻¹ ]

0 10 20 30 40 50 60 70 80 90 100

−30

−20

−10 0 10 20 30

k [ 2π/40000 km⁻¹ ]

Phase speed [ m/s ]

0 10 20 30 40 50 60 70 80 90 100

−30

−20

−10 0 10 20 30

k [ 2π/40000 km⁻¹ ]

Phase speed [ m/s ]

(26)

Dry basic state: congestus mode structure

Most unstable wave: k = 39, λ = 1026 km, c = 0 m/s

Depth [ km ]

0 4 8 12 16

0 384 769 1153 1538

−1 0 1

x [ km ]

qθ_eb θ_b

−1 0 1

M_d: Conv M_d: Env

(27)

Dry basic state: congestus mode budget

θ

eb

budget:

−1 0

1 DD

E Sfc

u

_b

budget:

−1

−0.5 0 0.5 1

PG E Drg

0 384 769 1153 1538

−1 0 1

x [ km ]

(28)

4. Conclusions

Outline

1. Background

Khouider–Majda multicloud model

2. Multicloud model with boundary layer dynamics

Boundary layer equations

Coupling to free troposphere

Free troposphere equations

3. Waves and stability

Basic state

Moist basic state instabilities

Dry basic state instabilities

4. Conclusions

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4. Conclusions

Conclusions

Boundary layer dynamics in the multicloud model:

◮

Enhance instability of synoptic-scale moist gravity waves;

◮

Destabilize fast mesoscale gravity waves;

◮

Focus congestus instability at synoptic scales.

Details: Waite & Khouider 2009, J. Atmos. Sci. (in press)

Boundary layer dynamics in a simple model for convectively coupled gravity waves