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  • View in gallery

    Velocity components of the SLB circulation in the direction (vector) and in the direction (shaded).

  • View in gallery

    The distributions of (a) and (b) .

  • View in gallery

    The variation of (a) and (b) with .

  • View in gallery

    The distribution of (a) and (b) and the integral of (c) .

  • View in gallery

    (a) The relationship between the maximum value of and the height approaches its maximum value. (b) As in (a), but for . (c) As in (a), but for the integral of . The arrows denote the direction along which the vertical temperature decay scale increases.

  • View in gallery

    The near-surface temperature in the WRF ideal simulation for (a) the land breeze at 0300 UTC and (b) the sea breeze at 1200 UTC.

  • View in gallery

    The land-breeze circulation at 0300 UTC predicted by (a) the theory calculation and (b) the WRF Model simulation.

  • View in gallery

    The sea-breeze circulation at 1200 UTC predicted by (a) the theory calculation and (b) the WRF Model simulation.

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An Analytical Solution for Three-Dimensional Sea–Land Breeze

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  • 1 College of Global Change and Earth System Science, Beijing Normal University, and Joint Center for Global Change Studies, Beijing, China
  • 2 National Marine Environmental Forecasting Center, Beijing, China
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Abstract

Based on the hydrostatic, incompressible Boussinesq equations in the planetary boundary layer (PBL), the three-dimensional sea–land breeze (SLB) circulation has been elegantly expressed as functions of the surface temperature distribution. The horizontal distribution of the horizontal or vertical motion is determined by the first or second derivative of the surface temperature distribution. For symmetric land–sea and temperature distribution, the full strength of the sea breeze occurs inland but not at the coastline, and the maximum updraft associates with the heating center. Setting the temperature difference between land and sea (TDLS), which varies with the island size, there would exist an optimal island size corresponding to the strongest SLB circulation that weakens with both a larger and smaller island size. Each velocity component approaches a peak at a certain vertical level. Both the peak value and the corresponding vertical level link with the vertical scale of the surface temperature: the more significant the influence of the surface temperature vertically, the stronger the SLB circulation at a higher vertical level it induces. The Weather Research and Forecasting (WRF) Model's ideal simulation for the two-dimensional sea breeze is applied to verify the theory. Two cases, land breeze and sea breeze, further support the theory's results despite a certain slight discrepancy due to the highly simplified theoretical equations.

Denotes Open Access content.

Corresponding author address: Dr. YaoKun Li, College of Global Change and Earth System Science, Beijing Normal University, No. 19 Xinjiekouwai St., Haidian District, Beijing 100875, China. E-mail: liyaokun@bnu.edu.cn

Abstract

Based on the hydrostatic, incompressible Boussinesq equations in the planetary boundary layer (PBL), the three-dimensional sea–land breeze (SLB) circulation has been elegantly expressed as functions of the surface temperature distribution. The horizontal distribution of the horizontal or vertical motion is determined by the first or second derivative of the surface temperature distribution. For symmetric land–sea and temperature distribution, the full strength of the sea breeze occurs inland but not at the coastline, and the maximum updraft associates with the heating center. Setting the temperature difference between land and sea (TDLS), which varies with the island size, there would exist an optimal island size corresponding to the strongest SLB circulation that weakens with both a larger and smaller island size. Each velocity component approaches a peak at a certain vertical level. Both the peak value and the corresponding vertical level link with the vertical scale of the surface temperature: the more significant the influence of the surface temperature vertically, the stronger the SLB circulation at a higher vertical level it induces. The Weather Research and Forecasting (WRF) Model's ideal simulation for the two-dimensional sea breeze is applied to verify the theory. Two cases, land breeze and sea breeze, further support the theory's results despite a certain slight discrepancy due to the highly simplified theoretical equations.

Denotes Open Access content.

Corresponding author address: Dr. YaoKun Li, College of Global Change and Earth System Science, Beijing Normal University, No. 19 Xinjiekouwai St., Haidian District, Beijing 100875, China. E-mail: liyaokun@bnu.edu.cn

1. Introduction

The sea–land breeze (SLB), a common phenomenon mainly observed in the planetary boundary layer (PBL), is induced by the diurnal rhythms of the thermal contrast between land and sea. It is one of the oldest topics that meteorologists are interested in and has been studied extensively by numerous observational and numerical studies [for example, see review papers by Abbs and Physick (1992), Miller et al. (2003), and Crosman and Horel (2010)]. Moreover, analytical studies also make significant contributions for advanced understanding of the SLB. Early analytical modeling supposed that the SLB is mainly the balance between the pressure gradient caused by unequal heating and friction (Jeffreys 1922). The effect of Earth’s rotation was then taken into account in 1947. The deflecting force of Earth’s rotation (Coriolis force) was used to explain the influence of the geostrophic wind on the diurnal variations of the SLB (Haurwitz 1947) and to explain why the wind direction was not perpendicular to the coast when the SLB was blowing at full strength (Schmidt 1947). The force effect of a prescribed surface temperature function on the SLB circulation was then emphasized (Walsh 1974). The observed difference in the intensities of sea and land breezes (the sea breeze is generally stronger than the land breeze) was explained analytically by the time-varying diurnal variation of the stratification and the related variation of the eddy diffusion coefficients that were expected to give rise to a temporally asymmetrical circulation even when the surface forcing is symmetrical (Mak and Walsh 1976). The rotation of the direction of the SLB was also analytically investigated (Neumann 1977; Simpson 1996). Furthermore, the atmospheric response was discussed by the relative size of the diurnal heating and cooling frequency and the Coriolis parameter (Rotunno 1983). When , the atmospheric response is confined to within a distance of the coastline, and when , the atmospheric response is in the form of internal–inertial waves. However, giving nonperiodic forcing and realistic values of friction, only an elliptic solution (the case) could result (Dalu and Pielke 1989). The horizontal dimension of the SLB circulation was then investigated by Niino (1987). More recently, the linear theory of Rotunno (1983) had been extended and explored analytically by including the effect of background wind on the sea-breeze wave response (Qian et al. 2009) and the effect of a base-state thermal wind on the linear dynamics of the sea breeze (Drobinski et al. 2011). Besides, there are also many theoretical studies that solve the SLB circulation numerically (Estoque 1961; Fisher 1961; Neumann and Mahrer 1971; Mahrer and Pielke 1977; Yan and Anthes 1987; Arritt 1993; Feliks 2004; Antonelli and Rotunno 2007; Qian et al. 2012). We mainly focus on the analytical solution and do not introduce them explicitly.

For mathematical convenience, the SLB is mainly discussed in two-dimensional space—for example, in an xz plane. The streamfunction then could be introduced to simplify equations to an equation containing only one variable. The analytical solution then could be obtained to conduct a theoretical discussion. It would be much more complex when dealing with the three-dimensional situation. Zhang et al. (1999) made a good attempt. They reduced the three-dimensional equations to a complex partial differential equation, which is so complicated that they had to solve it by a numerical relaxation method. Jiang (2012a) extended the linear solutions in Rotunno (1983) and Qian et al. (2009) to the three-dimensional SLB perturbation induced by complex coastlines. He further discussed the complexity introduced by inversion and vertical stratification variation and wind shear (Jiang 2012b). The above studies applied a Fourier transform to combine the system equations into a single wave equation that could be solved analytically under some simple circumstances. Then, characteristics of the three-dimensional SLB could be discussed. However, when applying the three-dimensional models to complex coastlines, it is difficult to obtain analytical solutions, and numerical methods have to be used.

The SLB is a mesoscale response of the atmosphere to horizontal variations in surface heating (Walsh 1974) that associates with the temperature difference between land and sea (TDLS). Therefore, it should be written as a function of the temperature distribution even though the form might be complex. For example, Haurwitz (1947) got the expressions for and winds, but it is difficult to discuss in general terms owing to the complexity of the auxiliary constants. Based on the above idea, this paper tries to analytically identify the driving effect of the temperature distribution on the SLB circulation. Consistent with former researchers [e.g., see Rotunno (1983)], the horizontal acceleration in the PBL is induced by the imbalance of the pressure gradient, friction, and the Coriolis force, and the basic flow is neglected. Differently, to highlight the forcing effect of the surface temperature and to solve the equation analytically, the temperature equation is not solved, and the surface temperature is seen as an independent prescribed variable. The three-dimensional SLB circulation could eventually be derived as an elegant expression of the surface temperature distribution by just hypothesizing that the temperature exponentially decreases with height. The circulation characteristics are then discussed. The theoretical analysis could also be used in studies of the urban heat island and the oasis cold island effects.

2. Model framework

The SLB circulation mainly prevails in the lower atmosphere; therefore, Boussinesq equations in the PBL are used to conduct the theoretical analysis. The basic current is set at zero because the SLB circulation is mainly driven by the thermal contrast between land and sea. It is more significant in the weak basic current situation. Consider a cylindrical coordinate system in which the ground plane is at and increases upward; increases outward and increases counterclockwise. Incompressible and hydrostatic Boussinesq equations in the cylindrical coordinate system are
e1
e2
e3
e4
e5
where is the Coriolis parameter; is the acceleration of gravity; and are the eddy coefficients of viscosity and conduction; and are the reference density and temperature of the stationary atmospheric background; represent the deviations of temperature, pressure, and velocity components in directions from the stationary atmospheric background, respectively; is the buoyancy frequency; and is the heating function. Constant values are assigned to , , , , , and (). Equation (3) employs the hydrostatic approximation, which could describe the SLB circulation well in most cases (Walsh 1974; Niino 1987; Jiang 2012a). Equation (4) expresses the air mass conservation under the incompressible approximation. Applying the variable separation method, that is, [e.g., see Rotunno (1983)], where denotes one of the above unknowns and day−1 represents the diurnal variation, Eqs. (1), (2), and (5) could be written as
e6
e7
e8
The forms of Eqs. (3) and (4) have no variation because they do not explicitly contain the time partial derivative terms. It should be noted that has been replaced by in Eqs. (3), (4), (6), (7), and (8) for simplification.
Set to substitute Eq. (3) into Eqs. (6) and (7) to eliminate pressure :
e9
Suppose that when ,
e10
According to the above boundary condition, the integral of Eq. (9) is
e11
It could be written as
e12
where the operator and , and . The term could be called the revised Ekman elevation. Note that is not defined when in which case Eq. (11) could be directly solved by its twice integration with respect to . The boundary conditions of Eq. (12) are taken as
e13
where could be called the boundary condition operator.
The solution of Eq. (12) is
e14
where is the Green function. According to the nature of the Green function, it has the following form:
e15
where . To obtain the specific expression of Eq. (14), it is necessary to solve the specific expression of .
If the surface temperature exponentially decays aloft with a vertical scale of ; that is,
e16
where represents the surface temperature distribution. It is common practice to assume the heating function exponentially declines [e.g., see Rotunno (1983) and Jiang (2012a)]. Here we apply the same assumption on the temperature distribution to analytically solve Eq. (14). Based on the assumption, the expression of could be written as
e17

To guarantee the finite integral value in Eq. (17), the vertical scale must be larger than zero, which means the temperature would decrease and tend to zero with increasing height. Equation (16) might be a little rough for just setting the exponential decline in the temperature profile but neglecting the possibly existing fluctuations. However, the temperature would eventually tend to be zero when the height is high enough, especially for the SLB circulation, which is mainly driven by surface heating and only prevails in the lower 1–2 km. Therefore, it is reasonable to make such an assumption.

Substituting Eqs. (15) and (17) into Eq. (14), we get
e18
The horizontal velocity components then could be derived by separating Eq. (18) into real and imaginary parts; namely,
e19
e20
where and are functions governing the vertical distribution of the horizontal velocity components. Their forms are
e21
e22
For the SLB circulation, the latitudinal variation of the Coriolis parameter could be neglected. Therefore, the vertical velocity could be solved by integrating Eq. (4) under the boundary condition ; namely,
e23
where is the horizontal Laplace operator in a cylindrical coordinate system. By introducing the topographic lifting effect—that is, where , , and denote the topography distribution, the horizontal velocity components on the lower boundary, and the Hamilton operator in the cylindrical coordinate system—the vertical motion could be written as
e24
Equation (24) could be easily solved for both and are known variables.
Seen from Eqs. (19), (20), and (23), the three-dimensional SLB circulation is completely expressed as functions of the surface temperature distribution. They directly show the driving effect of the surface heating on the SLB circulation. Also, they offer a method to theoretically diagnose the SLB circulation through the surface temperature, which is very easy to observe. Giving the time-varying temperature distribution, the diurnal variation of the SLB circulation also could be analyzed. Then how should the temperature distribution be determined? Of course it could be obtained by solving the thermal equation Eq. (8). For example, we could assume that the vertical profile of the heating function satisfies the same principle as the vertical temperature profile—that is, , where is the horizontal structure of the heating function. Then an equation only containing one variable (the surface temperature) could be obtained by just substituting Eq. (23) into Eq. (8) to eliminate the vertical velocity; namely,
e25
Equation (25) is a Helmholtz equation of the surface temperature . Given known , could be obtained by solving Eq. (25) either analytically or numerically. However, in most cases, especially for complex coastlines, it is hard to solve Eq. (25) analytically. On the other hand, once the heating function is prescribed (no matter what pattern it is), the surface temperature distribution would be only determined by Eq. (25). Therefore, it is reasonable to directly specify the surface temperature distribution [equivalent to Eq. (25) already being solved] to solve the horizontal and vertical velocity components. Although this would nevertheless leave the buoyancy frequency undiscussed, it simplifies the derivation processes and benefits the elegant form of the three-dimensional SLB circulation.

3. Model results

Equations (19), (20), and (23) could be applied to any given temperature distribution representing different land and sea distributions. A typical land and sea distribution for the SLB circulation is an island surrounded by sea. Therefore, we specify a circular island whose center is located at the coordinate origin and coastline is at , where is the island radius. For land is smaller than , and for the sea is larger than . The three-dimensional SLB circulation then could be reduced to the two-dimensional situation—that is, in space due to symmetry. According to the fixed land and sea distribution, we set a similar symmetric surface temperature distribution
e26
where is the TDLS intensity, which is revised by a biharmonic term in the brackets to make sure is zero when . Here denotes the reference TDLS intensity and is given a constant value. When is greater than zero, it corresponds to the sea breeze and when is smaller than zero, it corresponds to the land breeze. The reference island radius is represented by , which could be specified by the actual land and sea distribution. We consider TDLS as a function of because when there is no island in the ocean (), TDLS should be zero and when the island is large enough, TDLS should be a finite value. We will explicitly explain why it takes such a form below. The term indicates the nondimensional horizontal surface temperature distribution. It shows that the temperature value would be unity when and when and close to zero when is large enough.
Then substituting Eq. (26) into Eqs. (19), (20), and (23) gives
e27
e28
e29
where and are the negative first and second derivatives of the horizontal surface temperature distribution. They characterize the horizontal distribution of the horizontal and vertical velocity components respectively. Setting s−1, m2 s−1, m s−2, K, km, m, and °C, the theoretical SLB circulation could be calculated in the plane (Fig. 1). The -direction wind component (maximum speed is about 2.5 m s−1) blows from sea to land at the lower layer and reverses its direction with slower speed at the upper layer, accompanying strong updrafts near the island center and weak downdrafts over the sea. The wind vector consists of a clockwise circulation centered at about 600-m height above the coastline (near ). The -direction wind component (shaded) blows counterclockwise (positive) with comparable intensity to . The horizontal wind vector does not blow perpendicular to the coast but deviates towards the right. The velocity components have significant horizontal variations, which are determined by the first and second derivatives of the surface temperature distribution. Both and possess the same horizontal variation characteristic governed by (see in Fig. 2a), which denotes the negative first derivative of the surface temperature distribution. The term increases and then decreases with increasing . Such a pattern means there exists a maximum value at a certain , which could be specified by solving the zero point of the first derivative of ; namely, . After simple derivation, the solution of the equation is . Therefore, would eventually decrease after first approaching to its maximum value at which is about 35 km for an of 50 km. This implies that the sea breeze at full strength would not just blow at the coastline but continues toward the island center. In the region where the surface temperature gradient is large enough, mainly dominates, and it would be very small when is larger than 120 km. The horizontal variation of the vertical velocity is regulated by (see Fig. 2b), which denotes the negative second derivative of the surface temperature distribution. Differently, gradually decreases to zero and to a minimum value and to infinitely close to zero when becomes larger and larger. The at which approaches its extreme values could be specified by solving the zero points of its first derivative; that is, . The solutions of the equation are and . Further analysis indicates approaches to its maximum and minimum values at the two zero points, respectively. The structure demonstrates that the updrafts would be strongest at the heating center and would gradually weaken to zero when increases to the coastline (), larger than which, it would change sign to become downdrafts, which also could approach its maximum value at another critical value (). This suggests that the updrafts occur on the land while downdrafts happen on the sea, and the coastline is a natural boundary between them. Also, it could be easily seen that the updraft strength is much greater than the downdraft strength although the total vertical transports are the same. Therefore, the sea breeze correlates with strong but narrow updrafts and weak and wide downdrafts.
Fig. 1.
Fig. 1.

Velocity components of the SLB circulation in the direction (vector) and in the direction (shaded).

Citation: Journal of the Atmospheric Sciences 73, 1; 10.1175/JAS-D-14-0329.1

Fig. 2.
Fig. 2.

The distributions of (a) and (b) .

Citation: Journal of the Atmospheric Sciences 73, 1; 10.1175/JAS-D-14-0329.1

As demonstrated before, and characterize the first and second derivatives of the surface temperature distribution. Once the surface temperature is specified, the horizontal structures of the horizontal and vertical velocity components are determined. The maximum values of and are approached and , respectively. Then, substituting the two values into and , we could get and . The intensity of the SLB circulation can be characterized by and . However, if is a constant value, the inverse proportions between , , and imply that the smaller the island size is, the stronger the SLB circulation is. This is obviously unreasonable because when there is no island in the ocean, there should be no SLB circulation. The irrationality could be overcome by setting the varying TDLS value so that it decreases with decreasing island size. Therefore, this is also the reason why Eq. (26) revises to zero for . To make sure the limits of and are zero when tends to zero, should be an infinitesimal of the variable , whose order must be larger than 2, the highest order of the infinitesimal variable on the denominator. Therefore, is taken as the biharmonic form. We also could appoint other forms such as . For example, Jiang (2012a) took a similar heating function. However it would make the surface temperature tend to infinity when tends to infinity if without any constraint. With the revised TDLS intensity, the SLB circulation would disappear for the no-land situation. Figure 3 further embodies such relationships. The peak of is at about = 60 km, which decreases with the increase or decrease of and becomes zero for enough small or large (Fig. 3a). Similarly, also has a peak at about = 50 km, which decreases for larger or smaller (Fig. 3b). This suggests that the horizontal and vertical motions would be strongest at certain , which could be called the optimal island size. The optimal island size for the horizontal motion is larger than that for the vertical motion. The dependence of the SLB strength on the island size illustrated above is in qualitative agreement with the previous numerical and theoretical studies (Neumann and Mahrer 1974; Abe and Yoshida 1982; Xian and Pielke 1991; Jiang 2012a). This further demonstrates the reasonability of the revised and the correctness of the theory. The optimal island scale is also determined by the island scale , and a smaller corresponds to a smaller optimal scale and vice versa (figure omitted).

Fig. 3.
Fig. 3.

The variation of (a) and (b) with .

Citation: Journal of the Atmospheric Sciences 73, 1; 10.1175/JAS-D-14-0329.1

On the other hand, the vertical variation of the SLB circulation is controlled by and . The expressions of and are a combination of the exponential and trigonometric functions. There are two extreme values for (Fig. 4a), which are located at about 200 and 900 m, respectively. The lower maximum value (larger than zero) is much larger than the upper minimum value (smaller than zero), suggesting a strong wind toward land in the lower layer and its weak return flow toward sea in the upper layer. It also could be seen intuitively from Fig. 1. Differently, is larger than zero and has only one maximum value at about the 400-m level (Fig. 4b), indicating the counterclockwise wind in the whole layer. Therefore, the horizontal wind constitutes a cyclonic circulation cell linking with the low pressure driven by the heating in the lower layer while an anticyclonic circulation cell exists in the upper layer. It is the integral of contributing to the variation of the vertical motion rather than itself. The integral of is larger than zero and has a maximum value near the 600-m level (Fig. 4c), implying the maximum updrafts occur near the 600-m level. In addition, the heights that and approach their maximum values are different. There is only one extra introduced parameter , which, as mentioned before, indicates the vertical scale by which the surface temperature exponentially decays aloft. Physically, a large means a more significant influence of the surface temperature on the SLB circulation. According to Eqs. (21) and (22), could affect the maximum values of , , and the integral of (, , and in abbreviated form), and the vertical level they approach to their maximum values. The terms , , and could be specified by solving the zero points of their first derivatives with respect to . However, no analytical solutions could be derived owing to the complex expressions. Figure 5 exhibits such relationships numerically. The arrows in Fig. 5 denote the direction along which increases. Both , , , and the corresponding vertical levels increase with increasing . This implies that the stronger vertical influence of the surface temperature would benefit a stronger SLB circulation occurring at a higher level. Also seen in Fig. 5, the same increments of , , and associate with different height increments. For example, the same increment of associates with the decreasing increment of the vertical level for increasing . This suggests that compared with the upward movement of the SLB circulation, the strong surface temperature influence would strengthen its intensity more. A reasonable SLB circulation strength needs a moderate that should be comparable to the revised Ekman elevation according to the derivation.

Fig. 4.
Fig. 4.

The distribution of (a) and (b) and the integral of (c) .

Citation: Journal of the Atmospheric Sciences 73, 1; 10.1175/JAS-D-14-0329.1

Fig. 5.
Fig. 5.

(a) The relationship between the maximum value of and the height approaches its maximum value. (b) As in (a), but for . (c) As in (a), but for the integral of . The arrows denote the direction along which the vertical temperature decay scale increases.

Citation: Journal of the Atmospheric Sciences 73, 1; 10.1175/JAS-D-14-0329.1

4. WRF Model verification

In the previous section, we elegantly express the three-dimensional SLB circulation as functions of the surface temperature distribution and then discuss the basic structure of the SLB circulation. In this section, we further apply the National Center for Atmospheric Research (NCAR) Advanced Weather Research and Forecasting (WRF) Model version 3.6.1 to verify the theoretical calculation. The equation set for the WRF Model is fully compressible, Eulerian, and nonhydrostatic with a run-time hydrostatic option. The model uses a terrain-following, hydrostatic-pressure vertical coordinate with the top of the model being a constant pressure surface. The horizontal grid is the Arakawa-C grid. The time integration scheme in the model uses the third-order Runge–Kutta scheme, and the spatial discretization employs second- to sixth-order schemes. The model supports both idealized and real-data applications with various lateral boundary condition options (Skamarock et al. 2008). It has been used extensively in idealized sea breeze simulations (Antonelli and Rotunno 2007; A. J. Gibbs 2008, unpublished manuscript; Crosman and Horel 2012; Steele et al. 2013).

The WRF Model ideal sea-breeze case allows the users to gain a physical understanding of the sea-breeze itself by only considering the most basic and necessary parameters required to reproduce the appropriate features (A. J. Gibbs 2008, unpublished manuscript). Details on the WRF Model configured as a two-dimensional sea-breeze simulation are given in Table 1. The model setup is for a 2D case with 202 grid points and 35 vertical levels (xz plane). The horizontal grid spacing is 2000 m, and the stretched vertical grid spacing is 85–130 m. The land occupies 50 grid points in the middle of the domain. The central longitude is taken as the prime meridian, which means the start hour is the local time as well as the universal time coordinated (UTC). There is a diurnal cycle, and the latitude and longitude are set for radiation to work. Full physics are employed, using the WRF double-moment (WDM) 5-class microphysics scheme, the Dudhia and Rapid Radiative Transfer Model (RRTM) radiation schemes, the revised Monin–Obukhov surface layer scheme, and the thermal diffusion land surface scheme. The ocean is initially set 7 K warmer than the land surface temperature. The simulation covers a 24-h time frame at 0000 UTC with no wind at the beginning.

Table 1.

WRF Model two-dimensional sea-breeze details (WRF namelist selection in parentheses).

Table 1.

The simulation depicts a clear picture of the diurnal evolution of the SLB circulation. A shallow weak land-breeze circulation is established over the coastline at approximately 0100 UTC. It breaks down and a sea-breeze circulation emerges simultaneously at 0900 UTC. The sea breeze strengthens and then weakens until 2100 UTC, when it is replaced by the land breeze. Two times (0300 and 1200 UTC) are selected to characterize the land-breeze and sea-breeze circulations, respectively. Figure 6 shows the near-surface temperature associated with the two cases. Because of the symmetry of land and sea distribution, the coordinate origin of the abscissa (represented by variable ) in Fig. 6 starts at the middle of the land and increases toward the sea, and the boundary between land and sea is at = 50 km. In the case for the land breeze, the weak TDLS intensity (about −1.8°C; see Fig. 6a) corresponds to a weak land-breeze circulation (maximum horizontal speed is about 0.3 m s−1; see in Fig. 7b). The land-breeze circulation mainly dominates in the range of 20–80 km. The wind blows from land to sea at the lower level, then ascends to the higher level over the sea, then returns to the land at the higher level, descends to the lower level on the land, and finally forms a complete anticlockwise land-breeze circulation cell that could vertically extend up to about a 1000-m level. The land breeze at the lower level is stronger than the return flow at the higher level. The center of the circulation cell sits at about the 500-m level above the boundary between the land and sea.

Fig. 6.
Fig. 6.

The near-surface temperature in the WRF ideal simulation for (a) the land breeze at 0300 UTC and (b) the sea breeze at 1200 UTC.

Citation: Journal of the Atmospheric Sciences 73, 1; 10.1175/JAS-D-14-0329.1

Fig. 7.
Fig. 7.

The land-breeze circulation at 0300 UTC predicted by (a) the theory calculation and (b) the WRF Model simulation.

Citation: Journal of the Atmospheric Sciences 73, 1; 10.1175/JAS-D-14-0329.1

We calculate the theoretical land breeze (see in Fig. 7a) driven by the temperature distribution in the WRF ideal simulation. It could be seen that the analytical solution is consistent with the WRF Model results qualitatively. For example, an analogical anticlockwise circulation cell horizontally ranges from about 40 to 60 km where the temperature gradient is obvious and vertically extends up to about 1000 m. The downdraft and updraft branches dominate on land and sea, respectively. The land breeze occurs at the lower level with the intensity about 0.3 m s−1, and the return flow at the higher level is weak. Of course, there exists a certain discrepancy between the theoretical and the WRF Model results. For example, the analytical land breeze is slightly stronger than its WRF Model counterpart but with a narrower spatial extent.

The situation is similar in the case for the sea-breeze circulation. TDLS decreases from 8° to 0°C with increasing (Fig. 6b). It varies gently inland and far off over the sea but changes sharply near the boundary between the land and sea. The circulation pattern associated with the temperature distribution in the WRF ideal simulation indicates a typical sea breeze (Fig. 8b). Updrafts induced by the warmer land are compensated by the downdrafts over the cooler sea, and the wind blows from sea to land at the lower level and reverses its direction at the upper level. A clockwise sea-breeze circulation centered near the 750-m level above = 40 km then forms. The WRF Model also predicts a sea-breeze front (SBF) denoting the boundary between the cooler and warmer air masses. It is located at about = 35 km, where the sea breeze weakens sharply towards inland areas. The analytical sea-breeze circulation (Fig. 8a) calculated by the WRF Model temperature distribution shows good consistence with its WRF Model counterpart. For example, an analogical circulation cell with almost equivalent intensity and spatial scope dominates near the boundary of land and sea. The circulation cell center also occurs at a higher level above the land. However, the theoretical solution predicts neither the existence of the SBF nor the asymmetry of the motion about the circulation cell center.

Fig. 8.
Fig. 8.

The sea-breeze circulation at 1200 UTC predicted by (a) the theory calculation and (b) the WRF Model simulation.

Citation: Journal of the Atmospheric Sciences 73, 1; 10.1175/JAS-D-14-0329.1

Based on the comparison with the WRF Model, it could be seen that the theory is correct, and it does have the ability to characterize the SLB circulation, although it does have certain imperfectness owing to the highly simplified equations for obtaining the analytical solution.

5. Conclusions

The SLB circulation is an interesting phenomenon that has been extensively investigated either analytically or numerically. Recent theoretical progress has extended to the complete three-dimensional circulation in which the analytical solution could only be derived for some simple coastline cases. On the other hand, how to directly express the driving effect of the surface temperature on the SLB circulation is still an unsolved question that is worth further exploring. Therefore, we try to theoretically answer the question by applying the hydrostatic and incompressible Boussinesq equations under the stationary atmospheric background in the PBL. The time partial derivatives could be eliminated by introducing separation of variables. Then the horizontal motion could be reduced into a second-order differential equation, which could be solved analytically by just introducing a reasonable hypothesis that predicts that the surface temperature exponentially decays with height. Then the vertical motion could be solved analytically by integrating the incompressible continuity equation. Eventually, the theory elegantly exhibits the driving effect of the surface temperature on the three-dimensional SLB circulation.

According to the theory, the horizontal variations of the horizontal and vertical motions are determined by the first and second derivatives of the surface temperature distribution rather than the temperature itself. We calculate the SLB circulation in a simple circular island case (the island radius is set to 50 km) by prescribing that the horizontal surface temperature decays exponentially. The maximum horizontal speed occurs at , which is not located at the coastline but inland and decreases toward both land or sea. The maximum updrafts are at the center of the island (associated with the heating center) while the maximum downdrafts happen at ; the strong, narrow updrafts and the weak, wide downdrafts are bounded to the coastline. Because TDLS should be zero when there is no island present in the sea and be finite when the island is large enough, we choose the TDLS to vary with the island size to satisfy such common sense. This could lead to an optimal island radius associated with the strongest SLB circulation. The optimal island size for the SLB circulation is consistent with previous numerical and theoretical studies.

The vertical distributions of the three-dimensional velocity components are determined by the vertical scale of the surface temperature. The wind blows toward the sea at the lower level and toward land at the upper level. However, the wind blows clockwise in the entire layer. This indicates a cyclonic circulation at the lower level but an anticyclonic circulation at the upper level. The vertical distribution of the vertical motion is larger than zero, which means updrafts and downdrafts occupy the entire layer on land and sea, respectively. Also, each velocity component approaches its maximum value at a certain level. Both the vertical level and the maximum value increase with the increasing vertical scale of the surface temperature, which means that a stronger surface temperature would induce a stronger SLB circulation located at a higher level. However, the variation is uneven. The same increment of the maximum value corresponds to a large height increment when the vertical scale is small but a small height increment when the vertical scale is large. This suggests that with increasing influence of the surface temperature, the increment of the maximum value would gradually increase while the increment of the height corresponding to the maximum value would gradually decrease. Therefore, a stronger influence of the surface temperature would strengthen the circulation intensity more.

To further verify the theory, we apply the widely used WRF Model two-dimensional sea-breeze simulation to conduct an ideal experiment. An island with a size of 100 km is placed in the middle of the domain. We integrate the model for total 24 h from 0000 UTC and choose two times (0300 and 1200 UTC) to characterize the land breeze and sea breeze, respectively. At 0300 UTC, a weak land-breeze circulation is established near the boundary between the land and sea where the horizontal temperature gradient is significant. The downdrafts and updrafts occur on land and sea, respectively, with a corresponding wind blowing toward the sea at the lower level and toward land at the upper level. At 1200 UTC, a strong sea breeze with a reversed direction has replaced the weak land breeze. The colder sea air mass accompanying the sea breeze encounters the warmer land air mass to form an obvious SBF located inland. Using the near-surface temperature distribution in the WRF ideal simulation, the theory could give analogical circulation patterns corresponding to the land breeze and sea breeze, respectively. Both the horizontal scale and the intensity are consistent with the WRF Model results. Despite the fact that the theory does not predict the asymmetry of the circulation pattern and the SBF, it features the nature of the SLB circulation.

Acknowledgments

This paper is supported by 973 Program (2014CB953903) and the Fundamental Research Funds for the Central Universities (2013YB45). The authors greatly thank Dr. Zhaoming Liang for his warm help on the WRF Model compilation and simulation. The authors appreciate the useful comments from Editor Dr. Grabowski and from two reviewers.

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