Poster - Department of Mathematics

T HRESHOLD M ODELS FOR R AINFALL AND C ONVECTION :
D ETERMINISTIC VERSUS S TOCHASTIC T RIGGERS .
(1)
(1,2)
Scott Hottovy
Sam Stechmann
(1) University of Wisconsin Mathematics Department, (2) University of Wisconsin Department of Atmospheric &
Oceanic Sciences.
S TATISTICS OF THE M ODELS : F REQUENCY
Stationary Density of S1 (varying λ) vs D1
C ONVERGENCE OF M ODELS
The S2 model captures most statistics well, but the exact formulas are hard
to analyze. The D2 model has simple formulas and the statistics resemble
observations. For large transition rates λ, S2 is approximated by D2.
Stationary Density of S2 (varying λ) vs D2
0.08
0.09
0.07
0.08
0.06
0.07
0.05
Stationary Density
Water vapor plays a major role in the onset of convection in tropical regions. Recent
observations have shown that water vapor has a critical value, depending on temperature,
that signals a transition to convection [1]. Our study aims to answer the question: what
are the underlying mechanisms of the transition to convection? Specifically, we aim
to develop a simple mathematical model of water vapor that resembles the observational
statistics of critical phenomena.
Stationary Density
WATER VAPOR ’ S ROLE IN CONVECTION
0.06
0.05
0.04
0.03
0.04
Theorem: Let (qtλ , σtλ ) be the solution of SDE (S2) with initial conditions
(q0 , σ0 ) constant for every λ and let (qt , σt ) be the solution to SDE (D2)
with the same initial condition (q0 , σ0 ). Then
0.03
0.02
0.02
0.01
0.01
0
0
−0.01
0
20
40
60
Moisture [mm]
80
100
−0.01
0
Semi−log plot of Dry State Stationary Density
20
40
60
Moisture [mm]
80
"
100
lim E
Semi−log plot of Wet State Stationary Density
λ→∞
−2
−2
10
−4
10
−5
10
−6
To model column water vapor qt as a function of time t, we use a simple Stochastic
Differential Equation (SDE). The SDE changes dynamics depending on the state of σt ∈
{0, 1}.
mdt + D0 dWt , σt = 0
dqt =
D1 > D0 > 0
−rdt + D1 dWt , σt = 1
Transition Rate [per hr]
qc
In Dry state
Moisture [mm]
Moisture [mm]
In Wet state
Time [hrs]
S1
Stochastic Trigger, 1 Threshold
50
60
Moisture, q
70
80
S TATISTICS : P RECIP. M EAN AND VARIANCE
The simplicity of the models allows exact formulas for all of the statistics.
Mean Precipitation S1 (varying λ) vs D1
Mean Precipitation for S2 (varying λ) vs D2
Switch to Wet
state
qc
No change in
dynamics
qnp
Switch to Dry
state
In Dry state
Switch to Wet
state at random
time
Moisture [unitless]
Moisture [unitless]
Preciptation Variance for S1 (varying λ)
Precipitation Variance For S2 and D2
qc
No change in
dynamics
qc
qnp
Time [hrs]
Switch to Dry state at
random time
C ONCLUSIONS
• The onset of convection is well modeled by a simple SDE. The SDE
must have either a deterministic trigger with two thresholds or a
stochastic trigger to capture the relevant statistics.
• The deterministic trigger model is preferable to work with because
formulas are easier to solve exactly. The D2 model still captures the
observational statistics.
• The D2 model is a good approximation to the stochastic trigger model
because the stochastic model converges to the deterministic trigger
model for large transition rates λ.
References:
[1] J.D. Neelin, O. Peters, K. Hales: The transition to strong convection. J.
Atmos. Sci., 66, 2367-2384, doi:10.1175/2009JAS2962.1 (2009).
Moisture [unitless]
Switch to Wet state at
random time
The main idea: Two types of Error. 1) ξ from delay in jumping time, and 2)
ζ accruing “catch up” error, can be made small for λ 1 .
• The stochastic trigger captures important statistics better than the deterministic trigger.
Time [hrs]
S2
Stochastic Trigger, 2 Thresholds
Moisture [mm]
Moisture [mm]
40
The cloud fraction does not change when the type of trigger or the number of thresholds
is changed.
Precipitation Variance [unitless]
D2
Deterministic Trigger, 2 Thresholds
In Wet state
Time [hrs]
30
qc
qnp
Moisture [mm]
D1
Deterministic Trigger, 1 Threshold
Switch to Dry state
at random time
80
λ
qc
Moisture [mm]
Switch to
Dry
70
r
E[σ = 1] =
m+r
0
Switch to
Wet
50
60
Moisture, q
Precip Variance
Transition Rate [per hr]
λ= ∞
0
40
10
Mean Precip [unitless]
λ= ∞
λ
−7
Mean Precipitation [unitless]
Transition Rates for the two Threshold models
−5
10
The water vapor pdfs are plotted above with D1 on the left and D2 on the right in solid red
(dry state) and blue (wet state). The corresponding stochastic triggers are plotted (S1 on
left, S2 on right) with various increasing values of the rate λ−1 = 4, .4, .04, .004 hours.
The cloud fraction is defined as the fraction of time, on average, that the cloud is raining.
To define σt , we define the transition rates in the plots below with one threshold (left) and
two thresholds (right).
Transition Rates for the one Threshold models
−4
10
10
−7
M ODELS
10
−6
10
Observational statistics of water vapor from [1]. The plots are of: (a) the mean, (b) variance of precipitation, (c) the normalized mean precipitation and the (d) frequency of precipitating points with respect to the water vapor value.
0≤t≤T
−3
Stationary Density
Stationary Density
−3
30
= 0.
10
10
10
sup |qtλ − qt |
2 #
Moisture
Further Reading: Stechmann, Neelin: A Stochastic Model for the Transition to Strong
Convection, J. Atmos. Sci. (2011)
Stechmann, Neelin: First-Passage Time Prototypes for Precipiation Statistics, J. Atmos.
Sci. (2014)
Acknowledgements:
SS was funded by ONR Young Investigator Program through grant ONR
N00014- 12-1-0744

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