Understanding large-scale atmospheric and oceanic flows with … · 2019. 11. 12. · Moist vs dry baroclinic instability (Dry) linear stability of the baroclinic jet Comparison of

Lecture 4:Applications to the

atmosphere,moist-convective

RSW

Introduction

Methodology

Constructing themodel

Limiting equationsand relation to theknown models

General propertiesof the modelConservation laws

Characteritics and fronts

Example: scattering of asimple wave on a moisturefront

Introducing evaporation

Moist vs drybaroclinicinstability(Dry) linear stability of thebaroclinic jet

Comparison of the evolutionof dry and moist instabilities

Conclusions

Literature

Understanding large-scale atmosphericand oceanic flows with layered rotating

shallow water models

V. Zeitlin

1Laboratory of Dynamical Meteorology, ENS, Paris, France

Non-homogeneous Fluids and Flows, Prague, August2012



RSW

Introduction

Methodology









Conclusions

Literature

PlanIntroductionMethodologyConstructing the modelLimiting equations and relation to the known modelsGeneral properties of the model

Conservation lawsCharacteritics and frontsExample: scattering of a simple wave on a moisturefrontIntroducing evaporation

Moist vs dry baroclinic instability(Dry) linear stability of the baroclinic jetComparison of the evolution of dry and moistinstabilities

ConclusionsLiterature



RSW

Introduction

Methodology









Conclusions

Literature

General problematics

Importance of moisture in the atmosphere: obvious.Influences large-scale dynamics via the latent heatrelease, due to condensation and precipitation.Atmospheric circulation modeling: equation of state of themoist air extremely complex. Discretization/averaging:problematic.Current parametrizations of precipitations and latent heatrelease:relaxation to the equilibrium (saturation) profile ofhumidity⇒ threshold effect⇒ essential nonlinearityConsequences: no linear limit; linear thinking: modaldecomposition, linear stability analysis, etc impossible⇒problems in quantifying predictability of moist - convectivedynamical systems.



RSW

Introduction

Methodology









Conclusions

Literature

Aims and method I

Aim:Understanding the influence of condensation and latentheat release upon large-scale dynamical processes

Reminder:I Simplest model for large-scale motions: rotating

shallow water.I Link with primitive equations: vertical averagingI Baroclinic effects: 2 (or more) layers.

Problem with this approach for moist air: averaging ofessentially nonlinear equation of state.



RSW

Introduction

Methodology









Conclusions

Literature

Aims and method II

Our approach

I Combine (standard) vertical averaging of primitiveequations between the isobaric surfaces with that ofLagrangian conservation of moist enthalpy

I Allow for convective fluxes (extra vertical velocity)across the isobars

I Link these fluxes to condensationI Use relaxation parametrization in terms of bulk

moisture in the layer for thecondensation/precipitation



RSW

Introduction

Methodology









Conclusions

Literature

Aims and method III

Advantages:

I Simplicity, qualitative analysis of basic phenomenastraightforward

I Fully nonlinear in the hydrodynamic sectorI Well-adapted for studying discontinuities, in

particular precipitation frontsI Efficient numerical tools available (finite-volume

codes for shallow water)I Various limits giving known modelsI Inclusion of topography (gentle or steep)

straightforward



RSW

Introduction

Methodology









Conclusions

Literature

Primitive equations in pseudo-heightcoordinates

ddt

v + fk × v = −∇φ

ddtθ = 0

∇ · v + ∂zw = 0

∂zφ = gθ

θ0

v = (u, v) and w - horizontal and vertical velocities,ddt = ∂t + v · ∇+ w∂z , f - Coriolis parameter, θ - potentialtemperature, φ - geopotential.



RSW

Introduction

Methodology









Conclusions

Literature

Moisture and moist enthalpy

Condensation turned off: conservation of specifichumidity of the air parcel:

ddt

q = 0.

Condensation turned on: θ and q equations acquiresource and sink. Yet the moist enthalpy θ+ L

cpq, where L -

latent heat release, cp - specific heat, is conserved forany air parcel on isobaric surfaces:

ddt

(θ +

Lcp

q)

= 0,



RSW

Introduction

Methodology









Conclusions

Literature

Vertical averaging with convective fluxes

3 material surfaces:

w0 =dz0

dt, w1 =

dz1

dt+ W1, w2 =

dz2

dt+ W2.

W

W2

1

θ

θ1

2

0

2z

z

z

1

Mean-field + constant mean θ →



RSW

Introduction

Methodology









Conclusions

Literature

Averaged momentum and mass conservationequations:

{∂tv1 + (v1 · ∇)v1 + fk × v1 = −∇φ(z1) + g θ1

θ0∇z1,

∂tv2 + (v2 · ∇)v2 + fk × v2 = −∇φ(z2) + g θ2θ0∇z2 + v1−v2

h2W1,{

∂th1 +∇ · (h1v1) = −W1,∂th2 +∇ · (h2v2) = +W1 −W2,



RSW

Introduction

Methodology









Conclusions

Literature

Linking convective fluxes to precipitation I

Bulk humidity: Qi =∫ zi

zi−1qdz. Precipitation sink:

∂tQi +∇ · (Qiv i) = −Pi .

In precipitating regions (Pi > 0), moisture is saturatedq(zi) = qs(zi) and the temperature of the air-massWidtdxdy convected due to the latent heat releaseθ(zi) + L

cpqs(zi), is the one of the upper layer: θi+1.

We assume "dry" stable background stratification:

θi+1 = θ(zi) +Lcp

q(zi) ≈ θi +Lcp

q(zi) > θi ,

with constant θ(zi) and q(zi).



RSW

Introduction

Methodology









Conclusions

Literature

Integrating the moist enthalpy we get

Wi = βiPi

with a positive-definite coefficient

βi =L

cp(θi+1 − θi)≈ 1

q(zi)> 0.

Last step: relaxation formula with relaxation time τ .

Pi =Qi −Qs

iτ

H(Qi −Qsi )

where H(.) is the Heaviside (step) function.



RSW

Introduction

Methodology









Conclusions

Literature

2-layer model with a dry upper layer

Vertical boundary conditions: upper surface isobaricz2 = const, geopotential at the bottom constant (ground)φ(z0) = const, Q2 = 0, Q1 = Q:

∂tv1 + (v1 · ∇)v1 + fk × v1 = −g∇(h1 + h2),

∂tv2 + (v2 · ∇)v2 + fk × v2 = −g∇(h1 + αh2) + v1−v2h2

βP,∂th1 +∇ · (h1v1) = −βP,∂th2 +∇ · (h2v2) = +βP,∂tQ +∇ · (Qv1) = −P, P = Q−Qs

τ H(Q −Qs)

α = θ2θ1

- stratification parameter.



RSW

Introduction

Methodology









Conclusions

Literature

Sketch of the model

θ

θ1

2h

h1

2W

P>0



RSW

Introduction

Methodology









Conclusions

Literature

Immediate relaxation limit

τ → 0, ⇒ P = −Qs∇ · v1 (Gill, 1982), and

∂tv1 + (v1 · ∇)v1 + fk × v1 = −g∇(h1 + h2),

∂tv2 + (v2 · ∇)v2 + fk × v2 = −g∇(h1 + αh2)

− v1 − v2

h2βQs∇ · v1,

∂th1 +∇ · (h1v1) = +βQs∇ · v1,

∂th2 +∇ · (h2v2) = −βQs∇ · v1,

humidity staying at the saturation value: Q = Qs.



RSW

Introduction

Methodology









Conclusions

Literature

Baroclinic reduction

Rewriting the model in terms of baroclinic and barotropicvelocities:

vbt =h1v1 + h2v2

h1 + h2, vbc = v1 − v2,

and linearizing in the hydrodynamic sector gives:∂tvbc + fk × vbc = −ge∇η,∂tη + He∇ · vbc = −βP,∂tQ + Qe∇ · vbc = −P,

,

where ge = g(α− 1), Qe = HeH1

Qs, η - perturbation of theinterface, He - equivalent height.Model first proposed by Gill (1982) and studied by Majdaet al (2004, 2006, 2008).



RSW

Introduction

Methodology









Conclusions

Literature

Quasigeostrophic limit

In the small Rossby number limit on the β-plane Lapeyre& Held (2004) model follows:

d (0)1dt

(∇2ψ1 + y − η1

D1) =

βPD1

,

d (0)2dt

(∇2ψ2 + y − η2

D2) = −βP

D2,

Here d (0)idt = ∂t + (v (0)

i · ∇), k × v (0)i = −∇ψi , Di = Hi

H0, and

ψ1,2 (geostrophic streamfunctions) are related to thefree-surface (η2) and interface (η1) perturbations as:

ψ1 = η1 + η2, ψ2 = η1 + αη2.



RSW

Introduction

Methodology









Conclusions

Literature

1-layer moist-convective RSW

In the limit H1/(H1 + H2)→ 0 the reduced-gravityone-layer moist-convective shallow water follows(Bouchut, Lambaerts, Lapeyre & Zeitlin, 2009):

∂tv1 + (v1 · ∇)v1 + fk × v1 = −∇η,∂tη +∇ · {v1 (1 + η)} = −βP,∂tQ +∇ · (Qv1) = −P,

(Nondimensional equations, η - free-surface perturbation)



RSW

Introduction

Methodology









Conclusions

Literature

Horizontal momentum

(∂t + v1 · ∇)(v1h1) + v1h1∇ · v1 + fk × (v1h1)

= −g∇h2

12− gh1∇h2 − v1βP,

(∂t + v2 · ∇)(v2h2) + v2h2∇ · v2 + fk × (v2h2)

= −αg∇h2

22− gh2∇h1 + v1βP,

Red: moist convection drag. Total momentum:v1h1 + v2h2 is not affected by convection.



RSW

Introduction

Methodology









Conclusions

Literature

Energy

Energy densities of the layers:{e1 = h1

v21

2 + g h21

2 ,

e2 = h2v2

22 + gh1h2 + αg h2

22 ,

For the total energy E =∫

dxdy(e1 + e2) we get:

∂tE = −∫

dx βP(

gh2(1− α) +(v1 − v2)2

2

).

1st term: production of PE (for stable stratification); 2ndterm destruction of KE.



RSW

Introduction

Methodology









Conclusions

Literature

Potential vorticity

(∂t + v1 · ∇)ζ1 + f

h1=ζ1 + f

h21

βP,

(∂t + v2 · ∇)ζ2 + f

h2= −ζ2 + f

h22

βP +kh2·{∇×

(v1 − v2

h2βP)}

,

where ζi = k · (∇× v i) = ∂xvi − ∂yui (i = 1,2)- relativevorticity.PV in each layer is not a Lagrangian invariant inprecipitating regions.



RSW

Introduction

Methodology









Conclusions

Literature

Moist enthalpy and moist PV

Moist enthalpy in the lower layer: m1 = h1 − βQ and isalways locally conserved:

∂tm1 +∇ · (m1v1) = 0.

Conservation of the moist enthalpy in the lower layerallows to derive a new Lagrangian invariant, the moist PV:

(∂t + v1 · ∇)ζ1 + f

m1= 0.



RSW

Introduction

Methodology









Conclusions

Literature

Quasilinear form and characteristic equations

1-d reduction: ∂y (...) = 0, ⇒ quasilinear system:

∂t f + A(f )∂x f = b(f ).

Characteristic equation: det(A− cI) = 0I "Dry" characteristic equation

F(c) ={

(u1 − c)2 − gh1

}{(u2 − c)2 − αgh2

}−gh1gh2 = 0,

I "Moist" characteristic equation (τ → 0)

Fm(c) = F(c) + ((u1 − u2)2 − (α− 1)gh2)gβQs = 0.



RSW

Introduction

Methodology









Conclusions

Literature

Characteristic velocities about the rest state

I "Dry" characteristics:

C± = g(H1 + αH2)1±√

∆

2,

I "Moist" characteristics:

Cm± = g(H1 + αH2)

1±√

∆m

2.

Here C = c2 and

∆ = 1− 4H1H2(α− 1)

(H1 + αH2)2 =(H1 − αH2)2 + 4H1H2

(H1 + αH2)2 .

∆m = ∆ +4(α− 1)βQsH2

(H1 + αH2)2 .



RSW

Introduction

Methodology









Conclusions

Literature

Moist vs dry characteristic velocities

cm is real for positive moist enthalpy of the lower layer inthe state of rest : M1 = H1 − βQs > 0, and

Cm− < C− <

g(H1 + αH2)

2< C+ < Cm

+ ,

for 0 < M1 < H1 ⇒ moist internal (mainly baroclinic)mode propagates slower than the dry one, consistentwith observations.



RSW

Introduction

Methodology









Conclusions

Literature

Discontinuities in dependent variables (norotation)

Rankine-Hugoniot (RH) conditions (immediaterelaxation):

−s[v1h1 + v2h2] + [u1v1h1 + u2v2h2] = 0,−s[m1] + [m1u1] = 0,−s[h2] + [h2u2 + βQsu1] = 0.

s - propagation speed of the discontinuity.Remark: mass conservation→ moist enthalpyconservation in the lower layer.Due to limxs→a limb→xs

∫ ba P = 0, P does not enter RH

conditions for u, v ,h.



RSW

Introduction

Methodology









Conclusions

Literature

Discontinuities in derivatives

RH conditions linearized about the rest state:{(s2 − C+)(s2 − C−)[∂xu1] = −(α− 1)gH2gβ[P],(s2 − Cm

+ )(s2 − Cm− )[∂xu1] = −s(α− 1)gH2gβ[∂xQ].

For a configuration where it rains at the right side of thediscontinuity, P− = 0 and P+ = −Qs∂xu1+ > 0 , thereexist five types of precipitation fronts:



RSW

Introduction

Methodology









Conclusions

Literature

Precipitation fronts

1. the dry external fronts,√

C+ < s <√

Cm+ ,

2. the dry internal subsonic fronts,√

Cm− < s <

√C−,

3. the moist internal subsonic fronts, −√

Cm− < s < 0,

4. the moist internal supersonic fronts,−√

C+ < s < −√

C−,5. the moist external fronts, s < −

√Cm

+ .

This result confirms previous studies within a linearbaroclinic model (Frierson et al, 2004).



RSW

Introduction

Methodology









Conclusions

Literature

Wave scattering on a moisture front: setting

Localized internal simple wave centred at xP = 2 andmoving eastward:

u1(x ,0) =

{σ(x − xP)2 + U0 if −

√U0σ ≤ x − xP ≤

√U0σ ,

0 otherwise,U0 = 0.01, σ = −1(1)

Stationary moisture front at xM = 5, saturated air at theeast, unsaturated at the west:

Q(x ,0) = Qs{1 + q0 tanh(x − xM)H(−x + xM)},q0 = 0.05.(2)

Strong downflow convergence in the lower layer→ P > 0near the moisture front.



RSW

Introduction

Methodology









Conclusions

Literature

Wave scattering on a moisture front:baroclinic velocity and moisture

x

t

ubc

1 2 3 4 5 6 7 8 90

1

2

3

4

5

6

7

8

9

10

0

2

4

6

8

10

12

x 10−3

x

t

Q

1 2 3 4 5 6 7 8 90

1

2

3

4

5

6

7

8

9

10

0.83

0.84

0.85

0.86

0.87

0.88

0.89



RSW

Introduction

Methodology









Conclusions

Literature

Wave scattering on a moisture front:condensation zone I

x

t

ri+

s1

s2

4.4 4.6 4.8 5 5.2 5.4

5

5.5

6

6.5

7

0

2

4

6

8

10

12

14x 10

−3

x

t

rmi+

4.4 4.6 4.8 5 5.2 5.4

5

5.5

6

6.5

7

0

0.005

0.01

0.015

0.02

0.025

0.03

x

t

ri−

4.4 4.6 4.8 5 5.2 5.4

5

5.5

6

6.5

7

0

0.5

1

1.5

2

2.5

3

3.5

x 10−3

x

t

rmi−

4.4 4.6 4.8 5 5.2 5.4

5

5.5

6

6.5

7

−15

−10

−5

0

5

x 10−3

Dry and moist internal Riemann invariants. s1,2 -precipitation fronts (dry subsonic and moist supersonic).



RSW

Introduction

Methodology









Conclusions

Literature

Characteristics and fronts in thecondensation zone

t

x

S1

Q<Qs Q=Qs

Cq

Q<Qs Q=Qs

P>0

Cm

S2

Cd

P=0P=0

S3



RSW

Introduction

Methodology









Conclusions

Literature

Evaporation and its parametrizations

In the presence of evaporation source E

∂tQ +∇ · (Qv1) = E − P

Hence:∂tm1 +∇ · (m1v1) = −βE

Simple parametrizations of E (may be combined):

I Relaxational: E = Q−QτE

H(m1), where Q - equilibriumvalue.

I Dynamic: E = αE |v1|H(m1)

m1 should stay positive (plays a role of static stability)



RSW

Introduction

Methodology









Conclusions

Literature

Baroclinic Bickley jet

Geostrophically balanced upper-layer jet on the f -plane.non-dimensional profiles of velocity and thiknessperturbations:

u1 = 0, η1 =1

α− 1tanh(y),

u2 = sech2(y), η2 =−1α− 1

tanh(y).

No deviation of the free surface: η1 + η2 = 0.Parameters: Ro = 0.1, Bu = 10 - typical for atmosphericjets.



RSW

Introduction

Methodology









Conclusions

Literature

−3 −2 −1 0 1 2 30

0.5

1

u i/U

y/Rd

−3 −2 −1 0 1 2 3−10

0

10

η i/H0

y/Rd

−3 −2 −1 0 1 2 3−1

0

1

∆PV

i/fH0−

1

y/Rd

Bickley jet: zonal velocity ui , thickness deviation ηi andPV anomaly. Lower (upper) layer: solid black (dashed

gray).



RSW

Introduction

Methodology









Conclusions

Literature

Linear stability diagram

0 0.5 1 1.5 2 2.50

0.2

0.4

k

c p

0 0.5 1 1.5 2 2.50

0.1

0.2

k

σ

Phase velocity (top) and growth rate (bottom)



RSW

Introduction

Methodology









Conclusions

Literature

The most unstable mode

0.5 1 1.5 2 2.5

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

ψ2

x/Rd

y/Rd

0.5 1 1.5 2 2.5

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

ψ1

x/Rd

y/Rd

Most unstable mode of the upper-layer Bickley jet.Upper(top) and lower (bottom) layer- geostrophic

streamfunctions and velocity (arrows) fields.



RSW

Introduction

Methodology









Conclusions

Literature

Early stages: evolution of moisture

5Ti

(a)

0.5 1 1.5 2 2.5−1.5

−1

−0.5

0

0.5

1

1.5

−3

−2

−1

0

1

2

3x 10

−4

20Ti

(b)

0.5 1 1.5 2 2.5−1.5

−1

−0.5

0

0.5

1

1.5

−8

−6

−4

−2

0

2

4

6

8

x 10−4

200Ti

(c)

0.5 1 1.5 2 2.5−1.5

−1

−0.5

0

0.5

1

1.5

−0.11

−0.1

−0.09

−0.08

−0.07

−0.06

−0.05

−0.04

−0.03

−0.02

−0.01

300Ti

(d)

0.5 1 1.5 2 2.5−1.5

−1

−0.5

0

0.5

1

1.5

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

Evolution of the moisture anomaly Q −Q0 withsuperimposed lower-layer velocity. Black contour:condensation zones.



RSW

Introduction

Methodology









Conclusions

Literature

Early stages: growth rates

0 100 200 300 400 500−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

t/Ti

σ/Ro

DryMoist

Red: moist, blue: dry simulations. ⇒Transient increase in the growth rate due to condensation.



RSW

Introduction

Methodology









Conclusions

Literature

Cyclone-anticyclone asymmetry

0 500 1000 1500 2000

−0.5

0

0.5

t/Ti

UPPER LAYER

<ζ23 >/

<ζ22 >3/2

0 500 1000 1500 20000

2

4

6

t/Ti

LOWER LAYER

<ζ13 >/

<ζ12 >3/2

Skewness of relative vorticity. Red: moist, blue: drysimulations.



RSW

Introduction

Methodology









Conclusions

Literature

How condensation enhances cyclones:1-layer modelFor Ro → 0 and Bu ∼ O(1), close to saturationψ ∼ q << 1:

(∂t + v (0) · ∇)[∇2ψ − ψ

]= βP, (3)

(∂t + v (0) · ∇)[q −Qs∇2ψ

]= −P, (4)

v (0) = (−∂yψ, ∂xψ) - geostrophic velocity, ψ = η + η, andq is moisture anomaly with respect to Qs.⇒ PV of the fluid columns which pass through theprecipitating regions increases. For τ → 0 q ≈ 0, and:

Qs(∂t + v (0) · ∇)[∇2ψ

]≈ Pτ→0 > 0, (5)

⇒ increase of geostrophic vorticity in the precipitationregions.



RSW

Introduction

Methodology









Conclusions

Literature

Dry vs moist simulations: evolution of relativevorticity

Dry

200Ti

(a)

0.5 1 1.5 2 2.5−1.5

−1

−0.5

0

0.5

1

1.5

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

250Ti

(c)

0.5 1 1.5 2 2.5−1.5

−1

−0.5

0

0.5

1

1.5

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

300Ti

(e)

0.5 1 1.5 2 2.5−1.5

−1

−0.5

0

0.5

1

1.5

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

350Ti

(g)

0.5 1 1.5 2 2.5−1.5

−1

−0.5

0

0.5

1

1.5

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

Moist

200Ti

(b)

0.5 1 1.5 2 2.5−1.5

−1

−0.5

0

0.5

1

1.5

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

250Ti

(d)

0.5 1 1.5 2 2.5−1.5

−1

−0.5

0

0.5

1

1.5

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

300Ti

(f)

0.5 1 1.5 2 2.5−1.5

−1

−0.5

0

0.5

1

1.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

350Ti

(h)

0.5 1 1.5 2 2.5−1.5

−1

−0.5

0

0.5

1

1.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

Lower layer: colors, upper layer: contours. Condensation:solid black.



RSW

Introduction

Methodology









Conclusions

Literature

Dry vs moist simulations: formation ofsecondary zonal jets at late stages

Dry

400Ti

0.5 1 1.5 2 2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

800Ti

0.5 1 1.5 2 2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

Moist

400Ti

0.5 1 1.5 2 2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

800Ti

0.5 1 1.5 2 2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15



RSW

Introduction

Methodology









Conclusions

Literature

Unbalanced (aheostrophic) motions:baroclinic divergence

Dry

200Ti

(a)

0.5 1 1.5 2 2.5−1.5

−1

−0.5

0

0.5

1

1.5

−3

−2

−1

0

1

2

3x 10

−3

250Ti

(c)

0.5 1 1.5 2 2.5−1.5

−1

−0.5

0

0.5

1

1.5

−8

−6

−4

−2

0

2

4

6

8

x 10−3

300Ti

(e)

0.5 1 1.5 2 2.5−1.5

−1

−0.5

0

0.5

1

1.5

−0.01

−0.005

0

0.005

0.01

350Ti

(g)

0.5 1 1.5 2 2.5−1.5

−1

−0.5

0

0.5

1

1.5

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

Moist

200Ti

(b)

0.5 1 1.5 2 2.5−1.5

−1

−0.5

0

0.5

1

1.5

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

250Ti

(d)

0.5 1 1.5 2 2.5−1.5

−1

−0.5

0

0.5

1

1.5

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

300Ti

(f)

0.5 1 1.5 2 2.5−1.5

−1

−0.5

0

0.5

1

1.5

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

350Ti

(h)

0.5 1 1.5 2 2.5−1.5

−1

−0.5

0

0.5

1

1.5

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1



RSW

Introduction

Methodology









Conclusions

Literature

Moist baroclinic instability in Nature



RSW

Introduction

Methodology









Conclusions

Literature

Conclusions 1

The modelI Physically and mathematically consistentI Simple, physics transparentI Efficient high-resolution numerical schemes availableI Benchmarks: good



RSW

Introduction

Methodology









Conclusions

Literature

Conclusions 2

Moist vs dry baroclinic instability

I local enhancement of the growth rate of themoist-convective instability at the precipitation onset,

I significant increase in intensity of ageostrophicmotions during the evolution of the moist instability,

I substantial cyclone - anticyclone asymmetry, whichdevelops due to the moist convection effects.

I substantial differences in the structure of zonal jetsresulting at the late stage of saturation.



RSW

Introduction

Methodology









Conclusions

Literature

References

Presentation based on:I Lambaerts J.; Lapeyre G.; Zeitlin V. and Bouchut F.

"Simplified two-layer models of precipitatingatmosphere and their properties" Phys. Fluids 23,046603, 2011.

I Lambaerts J.; Lapeyre G. and Zeitlin V. "Moist versusDry Baroclinic Instability in a Simplified Two-LayerAtmospheric Model with Condensation and LatentHeat Release" J. Atmos. Sci. 69, 1405-1426, 2012.

Documents

Understanding large-scale atmospheric and oceanic flows with … · 2019. 11. 12. · Moist vs dry baroclinic instability (Dry) linear stability of the baroclinic jet Comparison of