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Three-Dimensional Structure of Atmospheric Fronts and Precipitations
Bykov P.L. [email protected], Gordin V.A.
[email protected]
National Research University - Higher School of Economic;
Hydrometeocentre of Russia, Moscow
General information on atmospheric fronts
There are two kinds only of discontinuity for the system of ideal gas dynamics equations:
1) Shock - jump of pressure and normal (to the discontinuity surface) velocity (wind) component;
2) Tangential discontinuity – jump of temperature, density and tangential (to the discontinuity surface)
velocity (wind) component. An atmospheric front – second kind discontinuity.
Practical task: to construct (by using discrete grid) 3D surfaces of atmospheric fronts. Atmospheric
fronts can be connected with a local revolution in weather (in meteorological fields) as well as
precipitations.
Obstacles:
Real front is not a jump, but a zone; its wide: from 1 kilometer up to tens km. In the same time typical
distance between aerological stations is equal to hundreds or thousands km. The front’s surface is
inclined with respect of Earth’s surface. Higher part of the front can join to tropopause (like horizontal).
The front’s surface is not smooth. Breaks and intersections like letter T are possible.
The field of horizontal wind (is shown by the arrows), was measured by profiler during atmospheric
front movement over station Lindenberg (Germany) in variables t, z; [7]. Long arrows correspond to 5
m/s, and the short one - to 2.5. The wind’s direction was changed from south and south-west to northeast. The period of direction’s change on the opposite one for any vertical level is equal to 1-2 hours; the
normal (to front’s surface) wind’s component is very small. Vertical temporal geometry of the evolution
is complex - we cannot assume that the frontal surface will be good approximated by a plane. Also we
can conclude that the vertical resolution (25 levels) is sufficient for the dynamics description.
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Laplacian, Hessian and a front’s geometry
Classical “frontal parameter” is the Laplacian of pressure on the sea level ∆pS . It is equal to infinity (if
the media is ideal) in the point of the unknown curve γ , and by the approach we have chance to find the
“hidden” hollow. It is a support of a distribution (generalized function) ∆pS of horizontal coordinates.
However a) The media is not ideal. b) We have data on a discrete grid only. It is a grid of
meteorological stations or a result of an interpolation from the grid to a regular (e. g. longitude-latitude)
grid – so called objective analysis (OA), or a result of forecast.
Therefore we can say (instead of the infinity) about relatively large values of the Laplacian. Not
only the function pS ( x, y ) may be used for such analysis, but other scalar functions, e. g. geopotential,
too. This “Laplacian approach” is not perfect: the second derivative in the normal (to the unknown
frontal line) direction is large, but the second derivative in tangential direction takes usual values.
Therefore the first addend is better than the Laplacian in 2 times.
Proposal: we use as a frontal parameter the large eigenvalues Π (x,y) of Hesse matrix:
⎛ ∂ 2xx f ∂ 2xy f ⎞
2
λ − λ∆f − h = 0, h = det ⎜⎜ 2
⎟⎟ , Π ( x, y ) = max [ λ1 ( x, y ), λ2 ( x, y ) ] .
2
⎝ ∂ yx f ∂ yy f ⎠
The eigen-values can be interpreted as curvatures of the graph of the function f(x,y).
Computation by wind’s field of a front’s line on a baric level
G
The second classical frontal parameter is the rotor of a wind w( x, y ) = u , v :
ωz = ∂ x v − ∂ y u.
(1)
In the ideal media the derivative across an ideal front is infinite, as well as the derivative along the front
takes the usual values. In the viscous model the first one is not infinite, but large only. If we use the formula,
the quality of the separation has worsened.
Proposal: we use as a frontal parameter the large eigenvalues Ω( x, y ) (x,y) of the following symmetrical
matrix:
uy
(v y − u x ) / 2 ⎞
⎛
λ 2 − λω z − det W = 0, W = ⎜
⎟ , Ω( x, y ) = max [ λ1 ( x, y ), λ2 ( x, y ) ]
−vx
⎝ (v y − u x ) / 2
⎠
instead of ωz .
G
G
Lemma. Let us consider for any unit vector b = cos ϕ ,sin ϕ the component of the vector field w
G
G G
in the direction: α ( x, y, ϕ ) = w, b and its derivative in the direction b ⊥ = − sin ϕ , cos ϕ . Then the
(
)
maximum and minimum of α with respect to the argument ϕ are equal to the eigen-values of the
matrix W(x,y).
Input information for atmospheric fronts’ determination
The atmospheric fronts’ geometry is evaluated on standard baric levels according to data of the objective
analysis or of the forecast by model “COSMO” (Consortium for Small scale Modeling) by basic
meteorological fields on regular grid with step 7km. The traditional predictors of the fronts are: 1) vertical
component of the wind’s vorticity; 2) Laplacian of pressure upon a sea level; 3) absolute value of the
temperature gradient. We use P1 , P2 , P3 (instead of them) - maximal eigenvalues of the following matrices:
a symmetric part of matrix Jacobi horizontal wind and the second horizontal derivatives (matrix Hesse)
geopotential (or pressure upon a sea level), correspondingly; P3 =| ∇ horT | . Here Pj = Pj ( p, x, y ) , where p –
pressure. We use high order approximations (fast Fourier transform and compact scheme approaches) for
every Pj calculation.
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Correlation function - definition & criterion for an atmospheric front
We calculate two kinds of correlation functions (CFs) for the basic meteorological fields
(geopotential, temperature, wind). Their CFs are functions of the distance between two points. Two
clusters of points’ pairs are used: A) the points are divided by a frontal zone and B) the points are not
divided by any frontal zone. Our algorithm must provide a maximal difference in L2 -norm (we
maximize the norm difference with respect to the functions f1 , f 2 , f 3 ) between these two CFs, and
(after the optimization) it provides the significant difference; see Fig.2.
Figure 2. Auto-CFs for basic atmospheric fields at the pressure level 400 hPa for optimal f1 , f 2 , f 3 . A
– for common air masse (green line), B – for the pair of points which are separated by an atmospheric
front (blue line) and pool two cases (red line). In statistics, involving A) 349659 and B) 284177 pairs of
measurements on upper-air meteorological stations in the period November 2009 - September 2011
We demonstrate frontal surfaces, see Fig.3, based on COSMO grid. Usually we obtain frontal surface
that across all troposphere. However, sometimes we observe such surface near Earth only, or, on the
contrary, in higher troposphere only. The 3D-geometry of such surfaces is sometimes very sophisticate.
We compare our results with satellites photos; see Fig.3,4.
Below
Infrared spectrum ~ 925 gPa;
Vapor spectrum ~ 500gPa.
Time is writhen in the right upper corner of the relative photos.
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Figure 3. Satellite photos of cloudiness in the infrared spectrum and our frontal lines according to forecast at the same moment on the level 925gPa
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Figure 4. Satellite photos of cloudiness in the vapor spectrum and our frontal lines according to forecast at the same moment on the level 500gPa
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Fronts and precipitations
The frontal zones connect with strong precipitations. Let us consider a field of hour’s precipitations
received from geostationary satellites. We have the global filed (without Polar Regions) with step
0.045D × 0.045D . We use as predictors of precipitations i) our frontal parameter; ii) humidity. We take
into account the predictors from several p-levels and construct a complex “integral” function. We
demonstrate here the probability of precipitations more then 0.5mm/hour divided on the mean
probability with respect to all Earth surface.
We can conclude that for low p-levels only strong (red line) atmospheric fronts connect with
precipitations, but about 700hPa, both strong and middle fronts are essential.
Connection AF with precipitations' zones
The precipitation’s forecast is more difficult than temperature or pressure forecast. The information
about AFs structure can be useful for the goal. We have used here our technology of AFs’ construction
and satellite data about precipitations quantity that became available for last years. The data include a
significant noise which appears as a result of errors of observations interpretation. The thin spatial
structure of precipitations’ zones is a special study object for the forecast goals. We have obtained
satellite precipitation data for small squares 0.045º×0.045º for the band between 30°-70° North latitude
It is a result of NOAA (National Oceanic and Atmospheric Administration) monitoring. It is published
on the server ftp://ftp.star.nesdis.noaa.gov/pub/smcd/emb/f_f/hydroest/world/world/ We used for our
statistical evaluation about 40.7 millions of such hour precipitations data in
0.5D × 0.5D
squares for the
Marths 2010 - 2012 years. and averaged them by squares 0.5º×0.5º. Probability of precipitation is greater
than 0.5 mm/h is 7.12% and mean precipitation rate is 0.144 mm/h.
We have compared the mean values (with respect to the band) with the same values in thin zones near
strong and middle (according to our classification) AFs for various baric levels. It turned out, that not
only the distance from an AF is essential, but the direction of the AF’s movement. According to Figure
6, the precipitations before an AF are more probable and intensive, than after an AF. The influence of an
AF on the baric level 800 gPa is most essential for the precipitations. For the prediction's goals we
should form a complex predictor, that include the distance from an AF, the AF velocity and humidity for
various baric levels, etc. According to Figure 7, there are several local maxima of the precipitation rate
before an AF, i.e. usually there is a thin temporal-spatial precipitations structure near AF. We have
obtained more details for the pictures, a) when the statistics was separated on the cold and warm AFs; b)
when we add occluded AFs. We use the following coordinates for the Figure 7: relative humidity – as
vertical coordinate, and AF intensity – as horizontal. Mean precipitation rate per next 6 hours grows with
respect to the both parameters.
Therefore the information can be used for a practical precipitation forecast!
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Fig 5. a) The increase in the relative probabilities of more than 0.5 mm precipitations per last hour when
passing intermediate or strong AF at baric level p in moment t = 0. A value of 1 corresponds to the probability
of precipitation more than 0.5 mm/h at any point, which used to archive the same 7.12%. b) The increase
in the mean precipitation rate per last hour. A value of 1 corresponds to the mean precipitation rate at any
point, which used to archive the same 0.144 mm/h. Each point on the graphs corresponds to 5·105 cases.
Figure 6. The increase in the mean precipitation rate per next 6 hours after passing AF. Relative
humidity plotted on the vertical axis and AF’ intensity (IAF) on the horizontal axis. A value of
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1 corresponds 0.866 mm/6h. For each shown interval of values humidity in the statistic is included
1.4·106 cases for the IAF=1 and 1.2·104 cases for the IAF=8.
Refernces
1. O.A.Aldukhov, Ph.L.Bykov, V.A.Gordin. Large-scale Three-Dimensional Correlation Functions for
Earth Atmosphere. Yaroslavl Pedagogical Vestnik, Ser. «Nature Sci.», 2012, №4, pp.36-43 (Russian).
2. O.A.Aldukhov, V.A.Gordin. Three-Dimensional Correlation Functions of Basic Upper-Air
Parameters. Proceedings of Russian Academy of Science, Ser. Physics of Atmosphere and Ocean, 2001,
V.37, №1, pp. 3-23 (Russian), 1-20 (English).
3. Ph.L.Bykov, V.A.Gordin. 3D Objective Analysis of the Structure of Three-Dimensional Atmospheric
Fronts. Proceedings of Russian Academy of Science, Ser. Physics of Atmosphere and Ocean, 2012,
V.48, №2, pp. 172–188 (Russian), 152-168 (English).
4. Ph.L.Bykov, V.A.Gordin. On Statistical Connection between Atmospheric Fronts and Precipitations.
Proc. of Hydrometeorological Center of Russia. 2012 № 348 (in press).
5. V.A.Gordin. Mathematical Problems and Methods in Hydrodynamical Weather Forecasting. Amsterdam etc.: Gordon & Breach Publ. House, 2000, 842p.
6. V.A.Gordin. Mathematics, Computer, Weather Forecasting, and Other Mathematical Physics’
Scenarios. - Moscow, Fizmatlit, 2010, 2012 (in Russian), 736p.
7. J. Neissen, V. Görsdorf, and H. Steinhagen, Instruments and Observing Methods (WMO, Geneva,
1994), No. 57.
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