Large eddy simulations forterrestrial convective mixed layers
Junshi Ito(Meteorological Research Institute)Outline
LES studies:
1. Earth convection– Tropical cyclone

2. Terrestrial convective mixed layer
2.1 Dust devil
2.2 Development of parameterization:
Horizontal eddy diffusivity for numerical weather
prediction models

1. LES for entire tropical cyclone• “Japan metrological agency’s non-hydrostatic model” generally
used for operational regional weather prediction→ Used as LES with cloud physics– Horizontal resolution: dx=100 m– Domain: 2000km×2000km×27km
Idealize experiment– Grid number: 20000×20000×60– Doubly periodic horizontal boundaries– Horizontal homogenous surface (sea)– Initial condition: Tropical cyclone developed with dx=2 km isinterpolated to dx=100 m
conducted by 1/8
nodes of K-computer(〜 10000 nodes)

Result (Cloud water amount, 9→10 hr)
200km

Result (Cloud water amount, 9→10 hr)

Studies on micro-scale structures in boundary
layer is on going2.1 LES for dust devil
a. Sensitivity test on diurnal variation, surface
heat flux, and ambient wind (Ito et al., 2010)
b. Formation mechanism (Ito et al., 2011; Ito et al., 2013)
c. Population of simulated dust devils
d. Feedback between surface heat flux and dust
devil wind
LES study
Ito (2011)↓PIV of dust
devil (Ito andNiino 2014)

LES model
Assume typical convective mixed layer on the Earth••••

Basic equations: 3D, Boussinesq approximation
Sub-grid model: Smagorinsky model
Grid interval: uniform Δ= 50 〜 5 m
Domain: e.g. 4.5 km×4.5 km×3.6 km (90×90×60 grids) for Δ=50 m
1.8 km×1.8 km×1.6 km (360×360×320 grids) for Δ= 5 m• Boundary conditions– Surface• Momentum: Bulk-method or Free-slip• Heat: Prescribe Horizontally uniform forcing with diurnal
variation or constant– Lateral: Doubly-periodic– Top: Free-slip• Initial conditions: Stable stratification Γ(＝4.0K/km)• Time integration: 0.2s stepSimulated dust devils
12:00~12:30, Δ=5m, vertical vorticity ζ>0.25s-1, ζ<-0.25s-1
Thick yellow at the bottom: updraft regions

a. Sensitivity test on diurnal variation,
surface heat flux, and ambient wind
• Impose diurnal-varying
surface heat flux
æt - 7ö
Q = Qmax × sin ç÷è 11 ø• Various sensitivity tests
Maximum of surface heat flux Qmax
Ambientwind

0.24K・m s-1

0.15K・m s-1

0.5m s-1

A1

B15m s-1

A2B215m s-1

A3B3

Favorable environment
Maximum vertical vorticity ζ(s-1) in horizontal sections at each height2500 m

A1

B1

A2

B2

A3

B3

0.24 K・m s-1

0.12
Larger surface heat flux

0.5m s-1

Weakerambient wind
5m s-115m s-1

0.15 K・m s-1

Structure in convective mixed layer
A1: Weaker ambient
wind, larger surface
heat flux

B3: Stronger ambient
wind, smaller surface
heat flux

Vertical velocity
in horizontal
section @14:00,
z=50m

Cellular convection

Roll convection

Favorable for dust devils
Consistent with observation (Hess and Spillane, 1990)

b. Formation mechanism
LES (e.g. Kanak 2005) have shown dust devils canoccur even in horizontally homogeneousconvective mixed layer without ambient wind
→ Formation mechanism?
No quantitative study on formation mechanism
has been presented

Vorticity analysis results in inconsistent between
backward-trajectories due to active CBL turbulence...
→ Circulation analysis (Ito et al., 2013)

Circulation analysis
Backward-trajectories
Flow
A material surface (MS) is
tracked with multiple
Trackbackward-trajectories

Circulation in MS

Fluid particles are distributed at
the core of dust devil
Change of the Γ: dG = F + B
ddt
Fd : Turbulent mixing
B : Baroclinic production

G = ò Sw × nds
Shrink of MS ⇔ Stretchingof vorticity
Tilting of MS
vorticity⇔ Tilting ofMaterial surface (MS) in a dust devil
Focusing on the strongest dust devil inLES (dx=5m, Free-slip)Patch

A MS of horizontal 20m×20m square at z=7.5m
is placed in the dust devil
The MS consisting of 40,000 triangle patches istracked with 20,000 backward-trajectories

Tracking of Material Surface
Shading:height (m)Results and discussion
Circulation ΓFactors tochange of ΓTurbulent mixing(horizontal, veritical）

TrackCBA
Baroclinic generation-120 s0s

Due to turbulent mixing
A: Γ decrease
B: Γ Increase
Dust devil forecast byobserving circulation?
C: Approximately constant→ Vertical vorticity stretching in convergencefrom a wide horizontal area is significant

c. Population of simulated
dust devils
LESOutput

Information of All pressure minima p’ < -5 Pa at
every time step
Track all simulated vortices automaticallythrough a day (0700−1700 LST);
only p’ <-10 pa are presented-10

Pressure depression (Pa)

Example: timeseries of p’ of eachtracked vortex
between 1330 and
1400 LST-10-15-20

Vortices merge-25-30-35-40-4013.5

13.6

13.7

13.8

Time (LST)

13.9

14

The number of Tracked vortices
LES: dx=20km, Domain: 4.2km×4.2km×4km
Maximum ofdiurnally varying
surface heat flux

The number ofvortices whosep’<-10 pa

0.24 K・ms-1

423

0.34 K・ms-1

757

0.5 K・ms-1

1728

Frequency of Intensity
Based on maximum -p’ of thorough its lifetime
10000

Qmax=0.24 K m/s
Qmax=0.34 K m/s
Qmax=0.5 K m/s

Frequency

1000∝p’ー4100

101
10203040

Maximum Pressure Depression (Pa)→Distributions appear to obey a power law

Lifetime vs Intensity
Maximum Pressure Depression (Pa)

9080
7060
504030
Qmax=0.24 K m/s
Qmax=0.34 K m/s
Qmax=0.5 K m/s

20100500

1000

1500

Lifetime (s)

2000

2500

Almost linear relationship: Lifetime ∝ ーp’Radius vs Intensity
Minimum pressure depression (Pa)40353025201510202530

3540

45

505560Radius of maximumwindthrough duration (m)Averagedradiusaveraged(m)→ weak positive relationship between radius
and intensity

Vortex merging vs Intensity
The number of merged tracked vortices (p’<−5 pa）
Maximum pressure depression (Pa)90

Qmax=0.24 K m/s
Qmax=0.34 K m/s
Qmax=0.5 K m/s

8070
60
50
40
30
20
100246810121416

The number of merged vortices• Some vortices merge more than 10 other vortices↔ Vortex merge is unnecessary for forming strong vortex

d. Feedback between surface heat flux and dust
devil wind
Surface heat flux Q:(LES so far) Horizontally homogeneous:

Q = Qf º 0.5× sin((t - 7) /11)

(In reality) Depends on near-surface wind speeds(i.e. Bulk method:)
U: Wind speed at lowest level

Possible positive feedback
Whirl wind induce locally large Q → Intensify dust devil →Stronger whirl wind → … (c.f. WISHE to develop Tropical cyclone）

Quasi-bulk
methodQ = Qf ×UU

Horizontal average remain the same<U>: horizontal average of U

Comparison of pressure
depression
5 samples with Quasi-

5 samples with usual LES(Pa)

bulk method00-20-20
Pressure Depression (Pa)Time seriesof minimumpressure
depression
at each time

Pressure Depression (Pa)

Homogenous Q

-40
-60
-80
-100
-120
-140

-40-60-80-100-120-140-160-160
910111213

Time (LST)1415169101112131415

Time (LST)The feedback between surface heat flux and whirl
wind may form stronger dust devils

162.2 LES to develop
parameterization
LES (Large Eddy Simulation)
resolve turbulent eddies inconvective mixed layers (CML)

Horizontal turbulent
diffusivity KhTurbulent flux

Vertical velocity in horizontal crosssection at the middle of CML↑

little investigated so far,but significant for highhorizontal resolution
Given<Horizontal turbulent flux> = - Kh × GradientPassive scalar to evaluate diffusivity
Passive scalar with uniform gradient ¶C = -1 as basic field¶y¶c¢¶c¢¶C ¶ æ ¶c¢ ö= -ui- uy+ çuq÷Equation of perturbation：¶t¶xi¶y ¶xi è ¶xi øyTurbulent flux vc’ on LES grid
PeriodicHeat flux<vc>=−<Horizontal turbulent diffusivity Kh> ×¶C ¶y< > : horizontal average over domain

LES model (almost the same as that for dust devil)
Basic equation

３-D, dry, and Boussinesq approximation

Sub-grid

Smagorinsky-Lilly model

Resolution

25 mDomain

18 km×18 km×5.0 km

Boundary
conditions

Lateral: doubly periodic
Bottom: Constant flux Q (Heat) + Friction
given by bulk method (Momentum)

Initial conditions

U=0m/s, Stable stratification（Γ=4.0K/km)

Time step0.2 sVertical profile of non-dimensional Kh(Ito et al. 2014)

1.6
From 5h (Q=0.2)
From 7h (Q=0.2)
From 12h (Q=0.2)
From 18h (Q=0.2)
From 8h (Q=0.1)

1.4

Height z/h

1.2

ScaledHeightz/h

1
Mean and standard deviation
σ during 1.5h

0.8
0.6
0.4

Possibly used forparameterization

0.2
000.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
Scaled Diffusion Coefficienth) /h)h/(w* /w
Non-dimensionalKhK(≡Kh*

Summary• LES for Earth: huge LES for entire tropical cyclone
was conducted• LES for terrestrial convective mixed layers– Dust devil• Favorable environment: weak ambient wind andstrong surface heat flux
• Formation Mechanism: Convergence ofpreexisted circulation over wide horizontal area
• Population is evaluated• Positive feedback between surface flux and wind
– Horizontal eddy diffusivity
Non-dimensional horizontal diffusivity
Kh~0.1

Scaling by w* for different standard
potential temperature θ0
Q=0.2Kms-1, Γ=0.004K/km, dx=50m, θ0=300K,400K, and 2000K (Difference between planets?)
w* º ( gQh q 0 ) 3
10.2

q0 = 300 K
q0 = 400 K
q0 = 2000 Kzmax (1/s)

0.15z max µ w*

0.1

0.05

00

0.5

11.5
w* (m/s)2

2.5

Convective velocity w* seems to be velocity scale→

What is length scale that depend little on time?

Wind speed estimated by PIV(Ito et al. 2014)• Particle Image Velocimetry (PIV) is performed• RGB difference between two successive time steps at thefront of the dust devil → Wind vectors in pixel/s• Stereoscope of the dust devil by two cameras and height ofobserver → 1 pixel = 13 cm (physical unit)0. PIV of a dust devil• Tucson, Arizona, 19/7/2012

3. Scaling of dust devils
CBL has self-similarity → Scaled by CBL height h,convective velocity w*, and so on (Deardroff 1970)
Vorticity of dust devil can be also scaled as〜(a velocity scale)/(a length scale)?

A natural choice:w* º ( gQh q 0 ) 31g: gravitational• A velocity scale: Convective velocity w*？ constant• A length scale: CBL height h？
→ Vorticity scale

w*µ h-3/2h

Inconsistent because vorticity scale
become small when CBL develop

Q: surface heat flux
θ0: standardtemperature

LES for exploring scaling
Domain： 4.5km×4.5km×3km; no ambient wind
Parameters to be changed• Surface heat flux Q: Constant (0.005, 0.1, 0.2,
or 0.3 K・m/s)• Grid resolution Δ: (50 → 6.25m)• Standard temperature θ0• Initial stratification Γ: 4, 8 K/m
CBL height h is also varied by timet; h increase as square root of t

h=2QtGScaling by w* for various Q
Correlation between w* and maximum verticalvorticity ζmax at z=75 m at each time-1

Q=0.05 Kms
Q=0.2 Kms -1-1
Q=0.3 Kms -1
Q=0.4 Kms

zmax (1/s)

0.2
0.15
0.1

z max µ w*

Data points in0.05
earlier time steps
are omitted
000.5

1

1.5
2w* (m/s)→ w* seems to be velocity scale

2.5

3

3.5

Vorticity〜Velocity/Length→ What is length scale that depend little on time?

Dependence on dx
Correlation between w* and ζmax; Q=0.2K・m/s, Γ=0.004K/m, Various dx
×0.5×0.5
ζmax:
maximum
vorticity at
z=75m

Const.×w*/Δ
×2×2

dx=50 m→12.5 m: halving dx doubles vorticity
dx=12.5 m → 6.25 m: vertical vorticity increase more
moderate

Scaling of circulation
Correlation between w* and circulation in the range of 50m by 50m
horizontal plane at z=75 m
1200
1000
2Circulation (m /s)Γmax∝ h×w*:
circulation can bescaled

dx=50mdx=25mdx=12.5m
dx=6.25m

800Almostconverged for
dx=6.25m

600400(Ito et al. 2013)

50m

200000.51w* (m/s)

1.52

Simulated dust devil vorticity may strongly depend onresolution dx

Effect of ambient rotation(Ito et al. 2012)
Time series of maximum Positive vertical vorticity and Negative verticalvorticity at z = 25 m with LES whose Δ = 50 m
LES with Coriolis effect f~10-3 1/s（10LES without Coriolis effecttimes as large as one at mid-latitude ）positive ζ

negative ζ

Difference of the order of 10-1s-1
Littlie difference betweenbetween positive and negative ζAmbient rotation 〜 10-3 s-1 → Difference in vertical vorticity at dust
devils 〜 10-1 s-1→ Convergence from wider horizontal area

6. Dust suspension• Include dust processes– Bulk formula for upward dust flux Fd at lowest level (Loosmore andHunt, 2000)
Fd≡3.6(u*)3, u*: frictional velocity– Gravitational settling– Prognostic equation of dust concentration c• No ambient wind
• dx=25m
3D image of dust concentration c at1500 LST
Shade: dust concentration
White circles: larger vertical vorticity

Dust amount in CBLHorizontally averaged dust concentration

1st day
Assumegravitational
settling at night Sunrise

2nd day
1.86 times larger
concentration→∞ dayc1+0.86c1+0.862c1+0.863c1+・・・+0.86∞c1→ 7.16c1

Sunset