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Studying coastal recirculation with a simplified analytical land-sea breeze model

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Studying coastal recirculation with a simplified analyticalland-sea breeze model
Ilan Levy,
1
Uri Dayan,
1
and Yitzhak Mahrer
2
Received 7 March 2007; revised 23 September 2007; accepted 23 October 2007; published 6 February 2008.
[
1
]
The diurnal cycle of the land and sea breeze is of high importance in determiningmany aspects of the living conditions in coastal areas. One of these aspects is the buildupof air pollutants concentrations due to air mass recirculation. In order to study the mainfactors governing the recirculation of an air mass under land-sea breeze conditions, asingle station characterization of the recirculation potential in an airshed is implementedusing a simplified land-sea breeze analytical model. The factors studied are latitude,ambient wind intensity, breeze intensity and friction. A sensitivity analysis performedreveals that the highest potential for coastal recirculation exists around latitude 30
°
withlow friction values. It is found that the combined effect of latitude, breeze intensity andthe meridional component of the ambient wind has the biggest influence on the modeland is responsible for 31% of its variance. Also, latitude and breeze intensity account eachfor about 20% of the variance. The recirculation model is found to be highly sensitive tolatitude, particularly in mid latitudes and to breeze intensity for weak breeze winds. The performance of the recirculation model is compared to 5-year measurements of recirculation at the East Mediterranean Sea during typical summer conditions. In spite of its rudimentary nature, the model does succeed in giving good quantitative measure of the recirculation, in the order of 0.3 on a scale of 0–1, very close to the observed values inthe region.
Citation:
Levy, I., U. Dayan, and Y. Mahrer (2008), Studying coastal recirculation with a simplified analytical land-sea breezemodel,
J. Geophys. Res.
,
113
, D03104, doi:10.1029/2007JD008628.
1. Introduction
[
2
] The meteorology of coastal regions in general and theLand-Sea Breeze (LSB) phenomenon in particular has beenstudied extensively during the last decades (see
NRC
[1992]and
Miller et al.
[2003] for extensive reviews), starting withobservational studies [e.g.,
Alpert et al.
, 1984;
Alpert and Rabinovich-Hadar
, 2003;
Yu and Wagner
, 1970], analyticalstudies [e.g.,
Dalu and Pielke
, 1989;
Defant
, 1951;
Haurwitz
, 1947;
Kusuda and Alpert
, 1983;
Neumann
,1977] and numerical models [e.g.,
Lu and Turco
, 1994,1995;
Pielke
, 1974]. The main reason for this interest is theLSB effect on local winds, air quality and convective activityin coastal regions [
Miller et al.
, 2003], where a large part of the world’s population lives, along with air pollutionemission sources and the associated environmental problemsarising [
NRC
, 1992].[
3
] The first analytical studies of the LSB [
Defant
, 1950;
Haurwitz
, 1947, 1959;
Pierson
, 1950;
Schmidt
, 1947] havealready shown that the main forces affecting the breeze arethe land-sea thermal gradient, Coriolis force, the large scale pressure gradient and friction. Later works have further studied the effect of these forces on more specific aspects of the LSB behavior, adding the effect of topography, staticstability and the diffusion of heat and momentum [
Simpson
,1994]. For example,
Estoque
[1962] studied the effect of large-scale wind on the sea breeze behavior.
Neumann
[1977] derived an expression for the time evolution of the breeze wind direction, and
Kusuda and Alpert
[1983] and
Alpert et al.
[1984] further studied the anti-clockwiserotation of the breeze.
Rotunno
[1983] and
Dalu and Pielke
[1989] have shown the effect of the Coriolis parameter andfriction on the extent of inland penetration of the sea breeze,while
Kitada et al.
[1986] analyzed the combined effect of the LSB and mountain-valley winds on air pollutants.[
4
] In addition to analytical works, numerical and obser-vational studies were also performed to measure the effect of these forces.
Zhong and Takle
[1993] and
Arritt
[1993]used numerical models to study the effects of large-scale background winds on the LSB circulation and found that onshore flows suppress the sea breeze as compared tooffshore flows.
Yoshikado
[1992] and
Martilli
[2003] havefound that the urban heat island accelerates the sea-breezeformation in the morning by the support of the urbancirculation, and also enhances vertical recirculation of air pollutants [
Martilli
, 2003]. The combined effect of moun-tains in the vicinity of the coastline and the LSB was foundto have a strong influence on pollutant transport, dependant
JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 113, D03104, doi:10.1029/2007JD008628, 2008
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for
Full Article
1
Department of Geography, The Hebrew University of Jerusalem,Jerusalem, Israel.
2
The Seagram Center for Soil and Water Sciences, The HebrewUniversity of Jerusalem, Jerusalem, Israel.Copyright 2008 by the American Geophysical Union.0148-0227/08/2007JD008628$09.00
D03104
1 of 13
on the mountains height and distance from the coast [
Lu and Turco
, 1994].[
5
] The effect of meteorological conditions on air pollu-tants’ concentrations is usually defined by the general termsof stagnation, recirculation and ventilation [
Allwine and Whiteman
, 1994]. Stagnation refers to events where thewind speed is very low and a polluted air mass remains over the same region for a period of time, and might absorbadditional primary pollutants that produce secondary pollu-tants [
AMS
, 2000]. Ventilation refers to the mass flux of clean air moving past an observer [
AMS
, 2000]. Ventilationoccurs when a polluted air mass is either being replaced bya fresh air mass or when the mixing depth increases and the pollutants are being diluted by turbulent mixing in a larger volume of air. Recirculation refers to events where thechange in wind direction causes an air mass to travel back to its source of srcin.[
6
] Air masses recirculation due to the diurnal rotation of the LSB winds plays an important role in determining manyaspects of coastal environments [
NRC
, 1992, chapter 7]. In particular, recirculation of polluted air masses may gravelyaffect the living conditions in cities residing along the coast.Indeed, many studies have shown that coastal cities expe-rience episodes of high air pollution levels due to recircu-lation, whether by field campaigns [e.g.,
Alper-Siman Tov et al.
, 1997;
Baumgardner et al.
, 2006;
Hanna et al.
, 1991;
Robinsohn et al.
, 1992] or by numerical models [e.g.,
Maand Lyons
, 2003;
Millan et al.
, 2002;
Oh et al.
, 2006].However, in spite of this abundance of evidence of therecirculation’s role in controlling air pollutants concentra-tions near the coast, as far as the authors are aware, noobjective methods have been proposed yet in order toquantify this phenomenon nor analyze the factors governingits occurrence and intensity.[
7
] Coastal recirculation could be separated into twodistinct processes: vertical and horizontal. Vertical recircu-lation involves the onshore winds near the surface, verticalascent at the sea breeze front, a return flow aloft a fewhundreds of meters above the ground and a descent to thesurface [
Hsu
, 1988]. Horizontal recirculation occurs due tothe diurnal clockwise (or anticlockwise) rotation of winddirection [
Miller et al.
, 2003].[
8
]
Allwine and Whiteman
[1994] (hereafter AW) have proposed an objective methodology to estimate atmosphericstagnation, recirculation and ventilation potential in anairshed, based on single-station integral measures of winddata. These quantities can be used to study the specific properties of the recirculation at the airshed, if applied onlong time series of wind observations for a sufficient number of meteorological stations in a given airshed.[
9
] This study makes use of AW’s method to studyhorizontal coastal recirculation in three cases: a) sinusoidalLSB oscillation, b) sinusoidal LSB oscillation with aconstant ambient wind-forcing and c) a solution for theequations of motion of
Haurwitz
[1947] (hereafter HW) for the LSB wind. The sensitivity of the recirculation model(i.e., implementation of AW’s recirculation parameter toHW’s equations of motion) is then evaluated by changingall input factors (i.e., ambient wind-forcing, LSB intensity,latitude and friction) over their entire range of feasiblevalues in order to study the main effect of each factor byitself and the effect of interactions between different factors.Finally, the model performance is validated and comparedto recirculation measurements at 18 air pollution monitoringstations located at three different urban airsheds along theIsraeli coastline on the East Mediterranean (EM) Sea, for a period of five years.
2. Analytical Formulation of Recirculation
2.1. Parameterization of Recirculation
[
10
] Given a series of N discrete observations of windvectors:
V
i
¼
u
i
þ
v
i
;
i
¼
1
;
2
;
. . .
;
N
ð
1
Þ
at a measuring site with an averaging interval of T hours(e.g., a 5-year series of hourly averaged wind observations),AW have suggested a single station measure for recircula-tion, stagnation and ventilation at the measuring site. On the basis of the wind data series, the authors have defined thediscrete integral quantities of resultant transport distance (L)that is the wind net vector displacement, wind run (S) that isthe wind scalar sum and recirculation parameter (R) basedon the ratio between L and S. All 3 parameters arecalculated for each time step t
i
as follows:
L
i
t
¼
T
X
i
þ
t
À
1
j
¼
i
V
j
¼
T
X
i
þ
t
À
1
j
¼
i
u
j
!
2
þ
X
i
þ
t
À
1
j
¼
i
v
j
!
2
2435
12
ð
2
Þ
S
i
t
¼
T
X
i
þ
t
À
1
j
¼
1
V
j
¼
T
X
i
þ
t
À
1
j
¼
i
u
2
j
þ
v
2
j
12
ð
3
Þ
R
i
t
1
À
L
i
t
S
i
t
ð
4
Þ
where
t
is the wind run time for integration (e.g., 24 h).[
11
] The integral quantities of L, S, and R are calculatedat every starting time step t
i
until the end time t
i+
t
À
1
. Theresultant transport distance L
i
is a measure of the net distance an air parcel had dislocated from the measuringsite after a period of
t
hours. S
i
is a measure of the totaldistance the parcel traveled in that time and provides anestimate of the stagnation. The combination of stagnant conditions (i.e., low S) and high recirculation is a measureof poor ventilation, whereas non-stagnant conditions withlow recirculation is a measure of good ventilation. L
i
divided by the wind run time
t
is the vector average windspeed, whereas S
i
divided by
t
is the scalar average windspeed. Figure 1 illustrates the definitions of S
i
and L
i
for thecases of high and low recirculation. The mathematicalsingularity of S = 0 requires a complete stagnation, meaningzero wind speed for a
t
period of time. In this case, L wouldalso be 0 and hence R is defined and has a value of 0 aswell.[
12
] It is important to note that these quantities are anexact measure of the parcel’s travel only for an idealhomogeneous wind field, i.e., when the wind observed at the measuring site is uniform throughout the region. Sincethis study implements these quantities for a LSB environ-
D03104
LEVY ET AL.: COASTAL RECIRCULATION ANALYTICAL MODEL2 of 13
D03104
ment, that is not the case. Therefore L, S, and R should only be regarded as a measure of the wind field in the vicinity of the measuring site. Only when calculated at multiple sites inan airshed can these quantities represent a larger region.Also, the progressive vector diagram (PVD) deduced fromwind speed and direction recorded at the measuring site isonly a coarse indication of the true path and thereforeshould not be considered as air mass trajectory.
2.2. Implementation
[
13
] The following discussion demonstrates the usage of AW’s parameterization of recirculation in 3 cases for agiven straight coastline with a north-south orientation,where the sea is to the west: a) LSB oscillation of asinusoidal nature, b) sinusoidal LSB oscillation with aconstant ambient wind-forcing and c) a more complex flowinvolving LSB, Coriolis force, large scale pressure gradient force and frictional force, following HW’s equations of motion.
2.2.1. Case (a) Sinusoidal LSB
[
14
] The simplest analytical example of the usage of AW’s parameterization of L, S, and R is a LSB with asinusoidal oscillation. Let us consider a north-south orientedcoastline where the land is to the east and an idealsinusoidal wind with a period of P = 24 h of the form:
u t
ðÞ¼À
A
cos
w
t
ð Þ ð
5
Þ
v t
ðÞ¼
A
sin
w
t
ð Þ
;
ð
6
Þ
where A is the breeze intensity (m s
À
1
) and
w
is the angular velocity of the Earth’s rotation (
w
= 2
p
/day). ApplyingAW’s method to equations (5) and (6) by calculating thedefinite integral for t = 0,
. . .
,
t
, yields analytical measuresfor L, S, and R of an air mass under the forcing of LSBsolely, as a function of
t
:
L
t
ð Þ¼
TA
w
2sin
wt
2
ð
7
Þ
S
t
ð Þ¼
TA
t
ð
8
Þ
R
t
ð Þ¼
1
À
1
wt
2sin
wt
2
:
ð
9
Þ
[
15
] Figure 2 shows the PVD of the wind run (a),resultant hodograph (b) and the dependence of R on
t
(c) for a sinusoidal wind with intensity of A = 1m sec
À
1
. Note that R(
t
) is equal to 1 every 24 h, as can be expectedfrom the diurnal cycle of the LSB in this case. Also, because the value of L is confined between 0
L
2TA/
w
and lim
t
!1
S
! 1
, then lim
t
!1
R
!
1. AW alsoshow that the local minima are described as:
R
n
¼
1
À
12 2
n
þ
1
ð Þ
;
n
¼
1
;
2
;
3
;
. . .
;
ð
10
Þ
at the corresponding discrete transport times of
t
n
=P(2n + 1)/2.
2.2.2. Case (b) Sinusoidal LSB With aConstant Ambient Wind:
[
16
] If we now add a constant ambient wind-forcing to thesinusoidal LSB in the form:
u t
ðÞ¼
u
g
À
A
cos
w
t
ð Þ
;
ð
11
Þ
v t
ðÞ¼
v
g
þ
A
sin
w
t
ð Þ ð
12
Þ
where u
g
and v
g
are constants, the resultant definite integralmeasures for L, S and R would be:
L
t
ð Þ¼
TA
w
u
g
wt
À
sin
wt
ð Þ
Â Ã
2
þ
v
g
wt
À
cos
wt
ð Þþ
1
Â Ã
2
n o
12
;
ð
13
Þ
S
t
ð Þ¼
TA
2
Z
t
0
2
v
g
sin
w
t
ð ÞÀ
2
u
g
cos
w
t
ð Þþ
u
2
g
þ
v
2
g
þ
1
h i
12
dt
;
ð
14
Þ
R
t
ð Þ¼
1
À
1
A
w
u
g
À
wt
À
sin
wt
ð Þ
Â Ã
2
þ
v
g
wt
À
cos
wt
ð Þþ
1
Â Ã
2
n o
12
R
t
0
2
v
g
sin
w
t
ð ÞÀ
2
u
g
cos
w
t
ð Þþ
u
2
g
þ
v
2
g
þ
1
h i
12
dt
:
ð
15
Þ
[
17
] The definite integral of S(
t
) in equations (14) and(15) can be found by means of Legendre’s incompleteelliptic integrals of the first and second kind. However, it
Figure 1.
Schematic illustration of the net vector dis- placement (L, dashed line) and progressive vector diagram(S, solid line) for high (top) and low (bottom) recirculation.The dot marks the monitoring site.
D03104
LEVY ET AL.: COASTAL RECIRCULATION ANALYTICAL MODEL3 of 13
D03104
is difficult to discuss the different parameters contribution torecirculation because of the complexity of a solution in thisform. Therefore a numerical approach is adopted.[
18
] As opposed to the first case where L was bounded between 0 and 2/
w
, now lim
t
!1
L
(
t
) =
1
, and therefore 0
lim
t
!1
R
(
t
)
1. In this case, the PVD has a diurnal breeze
Figure 2.
Progressive vector diagram (a), hodograph (b) and the recirculation parameter dependence ontransport time (
t
) (c) for pure sinusoidal wind with an intensity of 1 m s
À
1
.
Figure 3.
72 hrs progressive vector diagram (a), hodograph (b) and R(
t
) for different starting times between 00:00 and 23:00 (thin lines) and the average value for each
t
(bold) (c) for a sinusoidal LSB(A = 10 m s
À
2
) with constant ambient wind-forcing (u
g
= v
g
= 3 m s
À
1
).
D03104
LEVY ET AL.: COASTAL RECIRCULATION ANALYTICAL MODEL4 of 13
D03104
oscillation superimposed on the ambient wind direction(Figure 3a), so that the wind path never returns to its srcinand hence R < 1. Figure 3c presents R(
t
) for different starting times between 00:00 and 23:00 (thin lines) and theaveraged value for each
t
(bold line). The diurnal cycleintroduced by
w
in equations (11) and (12) causes R tohave a single value independent of the starting timewhenever
t
= 24n (n = 1, 2, 3
. . .
). In this case, theasymptotic value of R(
t
) is lim
t
!1
R
(
t
) =
R
24
(i.e., R
t
=24
), afunction of the relative intensities of the ambient wind andthe LSB.
2.2.3. Case (c)
Haurwitz
[1947] Equations of Motion
[
19
] A more realistic description of the LSB circulationwas presented by HW. Haurwitz provided a highly simpli-fied linear model for a north-south oriented shoreline wherethe land is to the west, that includes the Coriolis force (fu,fv), a large scale pressure gradient force (fu
g
, fv
g
), linear friction (ku, kv) and the diurnal oscillation of the local pressure gradient due to LSB with an amplitude A:
dudt
¼
f v
À
v
g
À Á
À
ku
À
A
p
À
12
A
cos
w
t
ð
16
Þ
dv dt
¼À
f u
À
u
g
À Á
À
kv
:
ð
17
Þ
[
20
] An approximation to the amplitude of the local LSBforcing is given by:
A
¼
gz T
a
@
T
@
X
;
ð
18
Þ
where g is the gravitation, z is the height where the pressuregradient due to land-sea temperature difference is no longer effective (i.e., the breeze layer height), T
a
is the mean air temperature of the breeze layer and
@
T/
@
x is the land-seatemperature gradient over the horizontal distance scale,taken as 60km. The constant part of the breeze (A/
p
) is ameasure to the relative intensities of the land and sea breezes. The Guldberg-Mohn friction coefficient, k [
Atkinson
, 1981, pp. 168], ranges from 2
Â
10
À
5
s
À
1
near the coast or over the ocean to 8
Â
10
À
5
s
À
1
over thecontinents [HW]. In spite of the shortcomings of this model,namely the omission of advection effects, vertical varia-tions, eddy viscosity, atmospheric stability, etc., therudimentary nature of these equations is sufficient for the purposes of studying the basics of coastal recirculation.[
21
] For consistency with the EM region studied in thiswork, where the land is to the east causing the sea breezeforce to act in the positive direction (i.e., from west to east),the LSB amplitude (A) in equation (18) was set as negative.The solution for u and v given by HW can be written in theform:
u t
ðÞ¼
A
1
À
B
1
sin
w
t
þ
C
1
ð Þ
;
ð
19
Þ
v t
ðÞ¼
A
2
À
B
2
cos
w
t
þ
C
2
ð Þ
;
ð
20
Þ
where:
A
1
¼
f
2
u
g
À
kfv
g
f
2
þ
k
2
À
A
p
k f
2
þ
k
2
ð
21
Þ
B
1
¼
A
2
k
2
þ
w
2
k
2
þ
w
2
À
f
2
ð Þ
2
þ
4
f
2
k
2
" #
12
ð
22
Þ
C
1
¼
arctg k
w
k
2
þ
w
2
þ
f
2
k
2
þ
w
2
À
f
2
ð
23
Þ
A
2
¼
f
2
v
g
þ
kfu
g
f
2
þ
k
2
þ
A
p
f f
2
þ
k
2
ð
24
Þ
B
2
¼
A
2
f
2
k
2
þ
w
2
À
f
2
ð Þ
2
þ
4
f
2
k
2
" #
12
ð
25
Þ
C
2
¼
arctg
2
k
ww
2
À
f
2
À
k
2
:
ð
26
Þ
u and v are now in the same form as for case b, except for the phase shift of the oscillating component, (i.e., C
1
andC
2
). The integral measures of L, S and R would now be:
L
t
ð Þ¼
T
w
A
1
wt
þ
B
1
cos
wt
þ
C
1
ð ÞÀ
B
1
cos
C
1
ð Þ½
2
n
þ
A
2
wt
À
B
2
sin
wt
þ
C
2
ð Þþ
B
2
sin
C
2
ð Þ½
2
o
12
;
ð
27
Þ
S
t
ð Þ¼
T
Z
t
0
A
1
À
B
1
sin
w
t
þ
C
1
ð Þ
2
þ
A
2
À
B
2
cos
w
t
þ
C
2
ð Þ½
2
n o
12
dt
;
ð
28
Þ
R
t
ð Þ¼
1
À
1
w
A
1
wt
þ
B
1
cos
wt
þ
C
1
ð ÞÀ
B
1
cos
C
1
ð Þ½
2
þ
A
2
wt
À
B
2
sin
wt
þ
C
2
ð Þþ
B
2
sin
C
2
ð Þ½
2
n o
12
R
t
0
A
1
À
B
1
sin
w
t
þ
C
1
ð Þð Þ
2
þ
A
2
À
B
2
cos
w
t
þ
C
2
ð Þð Þ
2
h i
12
dt
:
ð
29
Þ
[
22
] As in case b, due to the complexity of the definiteintegral of S(
t
) in equation (28), a numerical approach isadopted. During the reminder of this study, the implemen-tation of AW’s recirculation parameter to HW’s LSB modelwill be examined and referred to as the recirculation model.
3. Sensitivity Analysis and Model Performance
[
23
] In order to examine the impact of each of the fivevariables affecting the recirculation model, as formulated in
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LEVY ET AL.: COASTAL RECIRCULATION ANALYTICAL MODEL5 of 13
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