The stochastic ensemble of convective thermals (vortices), forming the fine structure of a turbulent convective atmospheric layer, is considered. The proposed ensemble model assumes all thermals in the mixed-layer to have the same determinate buoyancies and considers them as solid spheres of variable volumes. The values of radii and vertical velocities of the thermals are assumed random. The motion of the stochastic system of convective vortices is described by the nonlinear Langevin equation with a linear drift coefficient and a random force, whose structure is known for a system of Brownian particles. The probability density of the thermal ensemble in velocity phase space is shown to satisfy an associated K-form of the Fokker-Planck equation with variable coefficients. Maxwell velocity distribution of convective thermals is constructed as a steady-state solution of a simplified Fokker-Planck equation. The obtained Maxwell velocity distribution is shown to give a good approximation of experimental distributions in a turbulent convective mixed-layer.

1.
Yu. L.
Klimontovich
,
Turbulent Motion and the Structure of Chaos: New Approach to the Statistical Theory of Open Systems
(
Kluwer Academic Publisher
,
Dordrecht
,
1991
).
2.
N. G.
van Kampen
,
Stochastic Processes in Physics and Chemistry
, 3rd ed. (
Elsevier, North-Holland Physics Publishing
,
Amsterdam
,
2007
).
3.
C. W.
Gardiner
,
Stochastic Methods: A Handbook for the Natural and Social Sciences
, 4th ed. (
Springer-Verlag
,
Berlin-New York
,
2009
).
4.
P.
Hanggi
and
H.
Thomas
, “
Stochastic processes: Time evolution, symmetries and linear response
,”
Phys. Rep.
88
,
207
(
1982
).
5.
J. M.
Sancho
,
M.
San Miguel
, and
J.
Durr
, “
Adiabatic elimination for systems of Brownian particles with nonconstant damping coefficients
,”
J. Stat. Phys.
28
,
291
(
1982
).
6.
Yu. L.
Klimontovich
, “
Nonlinear Brownian motion
,”
Phys.-Usp.
37
,
737
(
1994
).
7.
G. K.
Batchelor
,
Introduction to Fluid Dynamics
(
Cambridge University Press
,
Cambridge
,
2000
).
8.
J. M.
Sancho
, “
Brownian colloidal particles: Ito, Stratonovich, or a different stochastic interpretation
,”
Phys. Rev. E
84
,
062102
(
2011
).
9.
C.
Kwon
and
P.
Ao
, “
Nonequilibrium steady state of a stochastic system driven by a nonlinear drift force
,”
Phys. Rev. E
84
,
061106
(
2011
).
10.
R. S.
Scorer
and
F. H.
Ludlam
, “
Bubble theory of penetrative convection
,”
Q. J. R. Meteorol. Soc.
79
,
94
(
1953
).
11.
H. E.
Huppert
and
J. S.
Turner
, “
Double-diffusive convection
,”
J. Fluid Mech.
106
,
299
(
1981
).
12.
A. N.
Vulfson
and
O. O.
Borodin
, “
Maxwell velocity distribution for a stochastic ensemble of thermals in a turbulent convective mixed-layer: Kinetic approach
,”
Proc. IUTAM
8
,
238
(
2013
).
13.
A. N.
Vulfson
and
O. O.
Borodin
, “
System of convective thermals as the generalized ensemble of Brownian particles
,”
Phys.-Usp.
59
,
109
(
2016
).
14.
G. K.
Batchelor
,
The Theory of Homogeneous Turbulence
(
Cambridge University Press
,
Cambridge
,
1953
).
15.
S. B.
Pope
,
Turbulent Flows
(
Cambridge University Press
,
Cambridge
,
2000
).
16.
G. S.
Golitsyn
,
Statistics and Dynamics of Natural Processes and Phenomena: Methods Tools and Results
(
Crasand
,
Moscow, Russia
,
2013
).
17.
B. H.
Burgess
,
D. G.
Dritschel
, and
R. K.
Scott
, “
Vortex scaling ranges in two-dimensional turbulence
,”
Phys. Fluids
29
,
111104
(
2017
).
18.
Z.
Chen
,
C.
Shu
, and
D.
Tan
, “
Three-dimensional simplified and unconditionally stable lattice Boltzmann method for incompressible isothermal and thermal flows
,”
Phys. Fluids
29
,
053601
(
2017
).
19.
C.
Nicolis
and
G.
Nicolis
, “
The fluctuation–dissipation theorem revisited: Beyond the Gaussian approximation
,”
J. Atmos. Sci.
72
,
2642
(
2015
).
20.
A. N.
Vulfson
and
O. O.
Borodin
, “
The Fokker-Planck equation and statistical characteristics of penetrative convection in the upper ocean
,”
Dokl. Earth Sci.
440
,
1287
(
2011
).
21.
Y.
Ogura
and
N. A.
Phillips
, “
Scale analysis of deep and shallow convection in the atmosphere
,”
J. Atmos. Sci.
19
,
173
(
1962
).
22.
A. N.
Vul’fson
, “
Equations of deep convection in a dry atmosphere
,”
Izv. Acad. Sci. USSR Atmos. Oceanic Phys.
17
,
646
(
1981
).
23.
J. W.
Deardorff
, “
Convective velocity and temperature scales for the unstable planetary boundary layer and for Rayleigh convection
,”
J. Atmos. Sci.
27
,
1211
(
1970
).
24.
O.
Zeman
and
J. L.
Lamley
, “
Modeling buoyancy-driven mixed layers
,”
J. Atmos. Sci.
33
,
1974
(
1976
).
25.
D. H.
Lenschow
,
J. C.
Wyngaard
, and
W. T.
Pennell
, “
Mean field and second-moment budgets in a baroclinic, convective boundary layer
,”
J. Atmos. Sci.
37
,
1313
(
1980
).
26.
J. C.
Kaimal
,
J. C.
Wyngaard
,
D. A.
Haugen
,
O. R.
Cote
,
Y.
Izumi
,
S. J.
Caughey
, and
C. J.
Readings
, “
Turbulence structure in the convective boundary layer
,”
J. Atmos. Sci.
33
,
2152
(
1976
).
27.
D. H.
Lenschow
,
M.
Lothon
,
S. D.
Mayor
,
P. P.
Sullivan
, and
G.
Canut
, “
A comparison of higher-order vertical velocity moments in the convective boundary layer from lidar with in situ measurements and large-eddy simulation
,”
Boundary-Layer Meteorol.
143
,
107
(
2012
).
28.
G. E.
Willis
and
J. W.
Deardorff
, “
A laboratory model of the unstable planetary boundary layer
,”
J. Atmos. Sci.
31
,
1297
(
1974
).
29.
J. W.
Deardorff
and
G. E.
Willis
, “
Further results from a laboratory model of the unstable planetary boundary layer
,”
Boundary-Layer Meteorol.
32
,
205
(
1985
).
30.
R. J.
Adrian
,
R. T. D. S.
Ferreira
, and
B.
Boberg
, “
Turbulent thermal convection in wide horizontal fluid layers
,”
Exp. Fluids
4
,
121
(
1986
).
31.
S. J.
Caughey
, in
Observed Characteristic of the Atmospheric Boundary Layer Atmospheric Turbulence and Air Pollution Modeling
, edited by
F. T. M.
Nieuwstadt
and
H.
van Dop
(
D. Reidel Publishing Company
,
Dordrecht
,
1982
), pp.
107
156
.
32.
A. M.
Obukhov
, “
Turbulence in thermally inhomogeneous atmosphere
,”
Boundary-Layer Meteorol.
2
,
7
(
1971
).
33.
C. H. B.
Priestly
,
Turbulent Transfer in the Lover Atmosphere
(
University Chicago Press
,
Chicago
,
1959
).
34.
A. N.
Vulfson
,
I. A.
Volodin
, and
O. O.
Borodin
, “
Local similarity theory and universal profiles of turbulent characteristics in the convective boundary layer
,”
Russ. Meteorol. Hydrol.
10
,
1
(
2004
).
35.
A. N.
Vulfson
and
O. O.
Borodin
, “
An ensemble of dynamically identical thermals and vertical profiles of turbulent moments in the convective surface layer of atmosphere
,”
Russ. Meteorol. Hydrol.
34
,
491
(
2009
).
36.
L.
Prandtl
, “
Meteorologische anwendungen der stromungslchre
,”
Beitr Phys. Atmos.
19
,
188
(
1932
).
37.
B. A.
Kader
and
A. M.
Yaglom
, “
Mean fields and fluctuation moments in unstably stratified turbulent boundary layers
,”
J. Fluid Mech.
212
,
637
(
1990
).
38.
N. I.
Vulfson
,
Convective Motions in a Free Atmosphere
(
Akad. Nauk SSSR
,
Moscow
,
1961
),
188
p. (Program for Scientific Translation, Jerusalem-Washington, 1964.)
39.
M. J.
Manton
, “
On the structure of convection
,”
Boundary-Layer Meteorol.
12
,
491
(
1977
).
40.
D. H.
Lenschow
and
P. L.
Stephens
, “
The role of thermals in the convective boundary layer
,”
Boundary-Layer Meteorol.
19
,
509
(
1980
).
41.
G. K.
Greenhut
and
S. J. S.
Khalsa
, “
Convective elements in the marine atmospheric boundary layer. Part I: Conditional sampling statistics
,”
J. Clim. Appl. Meteorol.
26
,
813
(
1987
).
42.
A. S.
Frish
and
J. A.
Businger
, “
A study of convective elements in the atmospheric surface layer
,”
Boundary-Layer Meteorol.
3
,
301
(
1973
).
43.
W. V. R.
Malkus
, “
The heat transport and spectrum of thermal turbulence
,”
Proc. R. Soc. A
225
,
196
(
1954
).
44.
W. V. R.
Malkus
, “
Outline of a theory of turbulent convection
,” in
Theory and Fundamental Research in Heat Transfer
, edited by
J. A.
Clark
(
Pergamon Press
,
Tarrytown
,
1963
), pp.
203
212
.
45.
A. A.
Townsend
, “
Temperature fluctuations over a heated horizontal surface
,”
J. Fluid Mech.
5
,
209
(
1959
).
46.
L.
Castillo
and
F.
Hussain
, “
The logarithmic and power law behaviors of the accelerating, turbulent thermal boundary layer
,”
Phys. Fluids
29
,
020718
(
2017
).
47.
R. S.
Scorer
,
Environmental Aerodynamics
(
Halsted Press
,
New York
,
1978
).
48.
J. S.
Turner
,
Buoyancy Effects in Fluids
(
Cambridge University Press
,
Cambridge
,
1995
).
49.
J.-I.
Yano
, “
Basic convective element: Bubble or plume? A historical review
,”
Atmos. Chem. Phys.
14
,
7019
(
2014
).
50.
J. S.
Turner
, “
Turbulent entrainment: The development of the entrainment assumption and its application to geophysical flows
,”
J. Fluid Mech.
173
,
431
(
1986
).
51.
J.
Simpson
and
V.
Wiggert
, “
Models of precipitating cumulus towers
,”
Mon. Weather Rev.
97
,
471
(
1969
).
52.
N. I.
Vulfson
and
L. M.
Levin
, “
Destruction of developing cumulus clouds by explosions
,”
Izv. Acad. Sci. USSR Atmos. Oceanic Phys.
8
,
156
(
1972
).
53.
O.
Sánchez
,
D. J.
Raymond
,
L.
Libersky
, and
A. G.
Petschek
, “
The development of thermals from rest
,”
J. Atmos. Sci.
46
,
2280
(
1989
).
54.
J. W.
Telford
, “
A theoretical solution to the motion of an atmospheric spherical vortex
,”
J. Atmos. Sci.
45
,
789
(
1988
).
55.
S.
Cheinet
, “
A multiple mass flux parameterization for the surface-generated convection. Part I: Dry plumes
,”
J. Atmos. Sci.
60
,
2313
(
2003
).
56.
C.
Jakob
and
A.
Siebesma
, “
A new subcloud model for mass-flux convection schemes: Influence on triggering, updraft properties, and model climate
,”
Mon. Weather Rev.
131
,
2765
(
2003
).
57.
P.
Siebesma
 et al, “
A large eddy simulation intercomparison study of shallow cumulus convection
,”
J. Atmos. Sci.
60
,
1201
(
2003
).
58.
B. R.
Morton
,
J. S.
Turner
, and
G. I.
Taylor
, “
Turbulent gravitational convection from maintained and instantaneous sources
,”
Proc. R. Soc. A
234
,
1
(
1956
).
59.
V.
Andreev
and
K.
Gannev
, “
Model of convective heat exchange due to isolated thermals in the atmospheric boundary layer
,”
Boundary-Layer Meteorol.
20
,
331
(
1981
).
60.
J.
Levine
, “
Spherical vortex theory of bubble–like motion cumulus clouds
,”
J. Meteorol.
16
,
653
(
1959
).
61.
C. P.
Wang
, “
Motion of a turbulent buoyant thermal in a calm stable stratified atmosphere
,”
Phys. Fluids
16
,
744
(
1973
).
62.
Yu. L.
Klimontovich
, “
Ito, Stratonovich and kinetic forms of stochastic equations
,”
Physica A
163
,
532
(
1990
).
63.
B.
Lindner
, “
The diffusion coefficient of nonlinear Brownian motion
,”
New J. Phys.
9
,
136
(
2007
).
64.
P.
Ao
,
C.
Kwon
, and
H.
Qian
, “
On the existence of potential landscape in the evolution of complex systems
,”
Complexity
12
,
19
(
2007
).
65.
P.
Ao
, “
Emerging of stochastic dynamical equalities and steady state thermodynamics from Darwinian dynamics
,”
Commun. Theor. Phys.
49
,
1073
(
2008
).
66.
A. N.
Vulfson
and
O. O.
Borodin
, “
On the stability of Maxwell distribution for a system of thermals in turbulent convective layers of fresh and salt water bodies
,”
Water Resour.
43
,
66
(
2016
).
67.
F.
Quintarelli
, “
A study of vertical velocity distributions in the planetary boundary layer
,”
Boundary-Layer Meteorol.
52
,
209
(
1990
).
68.
S. J.
Caughey
,
M.
Kitchen
, and
J. R.
Leighton
, “
Turbulence structure in convective boundary layers and implications for diffusion
,”
Boundary-Layer Meteorol.
25
,
345
(
1983
).
69.
V. Ya.
Fainberg
, “
Connection between the Fokker–Planck–Kolmogorov and nonlinear Langevin equations
,”
Theor. Math. Phys.
149
,
1710
(
2006
).
70.
J.
Dunkel
and
P.
Hanggi
, “
Theory of relativistic Brownian motion: The (1+1) dimensional case
,”
Phys. Rev. E
71
,
016124
(
2005
).
You do not currently have access to this content.