Evolution Characteristics of Urban Heat Island Circulation for Loess Tableland Valley Towns

Abstract

Urban heat island circulation (UHIC) determines the wind and thermal environments in urban areas. For Loess Tableland valley towns, the evolution characteristics of the UHIC over this negative terrain are not well understood, and therefore, it is important to investigate the evolution characteristics. A city-scale computational fluid dynamics (CSCFD) model is used, and simulation results are validated by the water tank experiment. The evolution process over such negative terrain can be divided into transient and quasi-steady stages, and in the transient stage, the airflow pattern evolves from thermal convection to city-scale closed circulation, while that in the quasi-steady stage is only city-scale closed circulation. In order to further reveal the characteristics of city-scale closed circulation, the sensitivities of different factors influencing the start time, outflow time, mixing height and heat island intensity are analyzed, and the most significant factors influencing these four parameters are urban heat flux, slope height, slope height, and potential temperature lapse rate, respectively. Finally, the dimensionless mixing height and heat island intensity for the valley town increase by 56.80% and 128.68%, respectively, compared to those for the flat city. This study provides guidance for the location and layout of built-up areas in the valley towns.

1. Introduction

Urban heat island circulation (UHIC) induced by the urban heat island is often formed under the stable stratification in no/weak background winds, and when the UHIC develops to the quasi-steady state, the typical characteristics of the UHIC includes the airflow at the lower level from suburbs to urban area (lower convergence inflow), airflow at the urban center from lower level to upper level (upward airflow), and airflow at the upper level from urban area to suburbs (upper divergence outflow) [1]. The UHIC not only affects the wind and thermal environments in urban areas, but also urban air quality [2,3]. There have been studies suggesting that the UHIC may pose a security threat because the lower convergent inflow can transport toxins released in suburban industrial buildings to urban areas [4]. With urban air pollution becoming more serious in recent years, the UHIC has been one of the focuses of research in air pollution meteorology to improve the livability of cities and the quality of life for residents [5]. Studies related to the UHIC can be categorized into two groups according to different terrains: one is the UHIC on the flat terrain and the other is the UHIC on the complex terrain [6].

The factors influencing the UHIC on the flat terrain include the city shape, stable stratification, background wind and so on [7]. The city shape mainly affects the symmetry of the UHIC. When the city shape is circular, the UHIC is symmetric on the vertical axis at the urban center [8]. When the city shape is an equilateral triangle (square), the UHIC is characterized as the lower convergent inflow along three (four) bisectors of the interior angle, upward airflow over the urban center, and upper divergent outflow perpendicular to three (four) sides [9]. When the city shape is rectangular, the upper divergent outflow is significantly affected by the aspect ratio of the long side to the short side. The study conducted by Fan et al. [9] shows that as the aspect ratio increases, the upper divergent outflows perpendicular to two short sides disappear, and those perpendicular to two long sides dominate. The stable stratification mainly affects the dimensionless mixing height of the UHIC, and the stronger the stable stratification is, the smaller the dimensionless mixing height is. The dimensionless mixing height, also referred to as the aspect ratio [10], is defined as the ratio of the mixing height at the urban center (z_ic) to the urban diameter (D). The study by Lu et al. [8] obtains an empirical formula between the dimensionless mixing height and the background buoyancy frequency: z_ic/D = 2.86 u_D/ND, where u_D is the horizontal velocity scale, N is the background buoyancy frequency and the stronger the stable stratification, the greater the background buoyancy frequency. The background wind mainly affects the structure of the UHIC [11]. When the 2.0 < R_v (R_v = u_D/U_b, U_b is the background wind speed), the UHIC is hardly affected by the background wind and is characterized by the upstream and downstream circulation organizations. When 0.5 < R_v < 2.0, the UHIC is distorted by the background wind and is characterized by the disappearance of upstream circulation organization. When R_v < 0.5, the UHIC is transformed into an urban plume by the background wind and is characterized by the disappearance of both upstream and downstream circulation organizations [7]. Therefore, the study of the UHIC can help to understand the formation mechanism and variation pattern of urban wind and thermal environments [12].

The UHIC on the complex terrain can interact with local winds to dominate urban ventilation [6]. Local winds mainly include the slope wind, valley wind, sea-land breeze and lake breeze [13]. For a city adjacent to mountains, the UHIC interacts with the slope wind. During the day, the upslope wind is formed. The upslope wind and the UHIC resist each other, which may cause the center of the UHIC to deviate from the urban center and change the structure of the UHIC [14]. During the night, the downslope wind is formed. On the one hand, the momentum of downslope winds can increase the velocity of lower convergent inflow, and thus promote the upward airflow over the urban center [15]; on the other hand, the cold energy of downslope winds can mitigate the heat island intensity [16]. For a city located in the valley, the UHIC interacts with the valley wind [17]. During the day, the upvalley wind is formed, and the downvalley wind is formed during the night. Vertical transport processes between the boundary layer and the free atmosphere can be increased considerably by the development of valley winds [18]. Therefore, the valley wind can obviously affect the UHIC characteristics; however, the interaction between the two has received less attention. For a city adjacent to the ocean, the UHIC interacts with the sea-land breeze. During the day, the sea breeze is formed, and the land breeze is formed at night. Before the arrival of the sea breeze, the UHIC for coastal cities may not develop enough, while the UHIC for inland cities can develop to the mature stage [19]. In that case, the former does not block the sea breeze, while the latter has a blockage effect on the sea breeze. Sensitivity experiments conducted by Wang et al. [20] reveal that either of the two urban areas, Beijing and Tianjin, can modify the penetration of the sea breeze. For a city adjacent to a lake, the UHIC interacts with the lake breeze. The full-scale simulation results by Liu et al. [21] reveal the synergy and competition between the UHIC and lake breeze. Therefore, the complexity of terrain can cause the particularity of the UHIC.

In summary, different terrains can cause different UHIC characteristics. For Loess Tableland valley towns [22], the town area is flanked by Loess Tablelands, as illustrated in Figure 1. The UHIC characteristics for the valley town have not been deeply explored, which seriously prevents some strategies proposed by urban planners and designers to mitigate the urban heat island effect and improve urban air quality [23]. The mountains on both sides of a city area can restrict the UHIC from developing in the horizontal direction, thus promoting the UHIC to expand in the vertical direction and eventually increasing the mixing height and heat island intensity [16,24]. Therefore, the Loess Tablelands on both sides of the town area also have a significant influence on the UHIC. This study will explore the UHIC for the Loess Tableland valley town and focus on the evolution characteristics of the UHIC. The findings will provide scientific guidance for the location and layout of built-up areas in the valley towns. This study proposes the numerical simulation method in detail in Section 2. In Section 3, the study presents simulation results and discusses reasons for the results. Some limitations are explained in Section 4. The main conclusions are drawn in Section 5.

2. Methods

2.1. Physical Model

The main body of the Loess Tableland valley town is long and narrow, which shows urban closure, spatial trend linearity and other characteristics [25]. Since the length of the valley floor in the y-direction is usually several dozen times that in the x-direction, the three-dimensional (3D) valley town can be simplified to a two-dimensional (2D) physical model [14], as shown in Figure 2. A full-scale numerical simulation for the 2D physical model is performed. In order to focus on the effect of this negative terrain of the Loess Tableland valley town on the evolution characteristics, the town area ignores the presence of a building complex, meaning that the town area is simplified as a flat plate that can heat the fluid. Although this simplification is ideal, several studies have demonstrated that such ideal models are fundamental to advancing the quantitative research of urban atmospheric dynamics on the complex terrain [16,26].

Since the physical model is symmetric about the urban centerline, only half of the physical model is simulated, and the symmetry boundary is used. The pressure outlet boundary is adapted for lateral boundaries. A damping layer with a thickness of 200 m is set at the top of the physical model for the purpose of avoiding the effect of spurious waves. The length of the top surface of the Loess Tableland is greater than 3.5D, which can reduce the influence of pressure outlet boundaries on the UHIC.

Table 1. Independent parameters of the UHIC for the valley town.

Number	Name	Symbol	Units	Dimensionless
1	Slope height	H	m	1
2	Instantaneous heat flux	Qi	K·m·s−1	1
3	Buoyant frequency	N	s−1	1
4	Urban length	D	m	Π1 = D/H
5	Suburban length	L	m	Π2 = L/H
6	Horizontal slope length	W	m	Π3 = W/H
7	Heating time	t	s	Π4 = tN
8	Buoyancy parameter	gβ	m·s−2·K−1	Π5 = gβQi/H2N3
9	Roughness height	zr	m	Π6 = zr/H
10	Kinematic viscosity	ν	m2·s−1	Π7 = ν/H2N
11	Thermal diffusivity	κ	m2·s−1	Π8 = κ/H2N

lists 11 independent parameters of the UHIC for the valley town. According to the Π theorem, eight Π terms can be obtained (Π₁~Π₈). This study focuses on the four Π terms, namely Π₁, Π₂, Π₃, and Π₄, because they are the core Π groups of the scale model experiment [27], and the regularities established using the core Π groups are applicable to both real atmospheres and water tank experiments. The core Π groups include seven variables, namely the slope height, instantaneous heat flux, buoyancy frequency, urban length, suburban length, horizontal slope length and heating time. The instantaneous heat flux can be expressed as Q_u/ρC_p, where Q_u is the urban heat flux, ρ is the air density, and C_p is the specific heat at constant pressure. The buoyancy frequency can be expressed as (gβ·∂T_p/∂z)^1/2, where g is the acceleration due to gravity, β is the thermal expansion coefficient, ∂T_p/∂z is the potential temperature lapse rate. The horizontal slope length can be expressed as H/tanθ, where θ is the slope angle. All seven variables can affect the evolution characteristics, and therefore orthogonal experiments are used to explore the effects of these variables on the evolution characteristics. As shown in

Table 2. Description of studied cases.

Case No.	Urban Length	Urban Heat Flux	Potential Temperature Lapse Rate	Suburban Length	Slope Height	Slope Angle
	D (m)	Qu (W/m2)	∂Tp/∂z (K/m)	L (m)	H (m)	θ (°)
1	700	50	0.003	250	200	28
2	1000	100	0.003	150	250	38
3	900	150	0.003	300	50	33
4	600	200	0.003	200	150	18
5	800	250	0.003	100	100	23
6	800	50	0.006	150	150	33
7	700	100	0.006	300	100	18
8	1000	150	0.006	200	200	23
9	900	200	0.006	100	250	28
10	600	250	0.006	250	50	38
11	600	50	0.009	300	250	23
12	800	100	0.009	200	50	28
13	700	150	0.009	100	150	38
14	1000	200	0.009	250	100	33
15	900	250	0.009	150	200	18
16	900	50	0.012	200	100	38
17	600	100	0.012	100	200	33
18	800	150	0.012	250	250	18
19	700	200	0.012	150	50	23
20	1000	250	0.012	300	150	28
21	1000	50	0.015	100	50	18
22	900	100	0.015	250	150	23
23	600	150	0.015	150	100	28
24	800	200	0.015	300	200	38
25	700	250	0.015	200	250	33

, except for the heating time, five levels are selected for each of the six variables, which are 600 m, 700 m, 800 m, 900 m, 1000 m for the urban length; 50 W/m², 100 W/m², 150 W/m², 200 W/m², 250 W/m² for the urban heat flux [7]; 0.003 K/m, 0.006 K/m, 0.009 K/m, 0.012 K/m, 0.015 K/m for the potential temperature lapse rate; 100 m, 150 m, 200 m, 250 m, 300 m for the suburban length; 50 m, 100 m, 150 m, 200 m, 250 m for the slope height; and 18°, 23°, 28°, 33°, 38° for the slope angle [28].

2.2. Numerical Model

The city-scale computational fluid dynamics (CSCFD) model has been successfully used to study the UHIC for plain cities [29]. The CSCFD model aims to resolve some of the incapability of the mesoscale model when it is used to model the wind flows around a building or in a street on a fine grid (e.g., 1 m) [30], and resolve the incapability of the microscale model when it is applied to the mesoscale atmospheric flows [31]. The governing equations of the CSCFD model are shown below, which are transformed from the governing equations of the traditional CFD microscale model by means of the vertical coordinate transformation method, and the new variables in the CSCFD model are represented by the subscript “n”.

The continuity equation

$${\displaystyle \frac{\partial u}{\partial x}}+{\displaystyle \frac{\partial v}{\partial y}}+{\displaystyle \frac{\partial w_n}{\partial z_{\mathrm{n}}}}=0$$

(1)

The momentum equation

$${\displaystyle \frac{\mathrm{d}\left({\rho }_0{\mathit{V}}_{\mathbf{n}}\right)}{\mathrm{d}t}}=-{\nabla }_{\mathrm{n}}{p}_{\mathrm{n}}+{\nabla }_{\mathrm{n}}\left({\mu }_{\mathrm{eff}}{\nabla }_{\mathrm{n}}{\mathit{V}}_{\mathbf{n}}\right)+{\rho }_0\beta \left(T_{\mathrm{n}}-T_0\right)g+{\mathit{F}}_{\mathit{C}}+{\mathit{F}}_{\mathrm{n}}$$

(2)

The energy equation

$${\displaystyle \frac{\mathrm{d}\left({\rho }_0{C}_p{T}_{\mathrm{n}}\right)}{\mathrm{d}t}}=\nabla \left(k_{\mathrm{eff}}\nabla T_{\mathrm{n}}\right)+{\rho }_0{Q}_{\mathrm{u}}+J{C}_p{\rho }_0{w}_{\mathrm{n}}\left(\mathsf{\Gamma }-{\displaystyle \frac{g}{C_p}}\right)$$

(3)

The turbulence equations

$${\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\left({\rho }_0{k}_{\mathrm{n}}\right)={\nabla }_{\mathrm{n}}\left(\left(\mu J_1+{\displaystyle \frac{{\mu }_t}{{\sigma }_k}}\right){\nabla }_{\mathrm{n}}{k}_{\mathrm{n}}\right)+G_{k\mathrm{n}}+G_{b\mathrm{n}}-{\rho }_0{\epsilon }_{\mathrm{n}}+S_{k\mathrm{n}}$$

(4)

$${\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\left({\rho }_0{\epsilon }_{\mathrm{n}}\right)={\nabla }_{\mathrm{n}}\left(\left(\mu J_1+{\displaystyle \frac{{\mu }_t}{{\sigma }_{\epsilon }}}\right){\nabla }_{\mathrm{n}}{\epsilon }_{\mathrm{n}}\right)+C_{1\epsilon }{\displaystyle \frac{{\epsilon }_{\mathrm{n}}}{k_{\mathrm{n}}}}\left(G_{k\mathrm{n}}+C_{3\epsilon }{G}_{b\mathrm{n}}\right)-C_{2\epsilon }{\rho }_0{\displaystyle \frac{{{\epsilon }_{\mathrm{n}}}^2}{k_{\mathrm{n}}}}+S_{\epsilon \mathrm{n}}$$

(5)

where u is the velocity in the x-direction, v is the velocity in the y-direction, w_n is the velocity in the z_n-direction, z_n is the transformed vertical coordinate, V_n (vector) is the transformed velocity, ρ₀ is the density, t is the time, p_n is the transformed pressure, μ_eff is the effective viscosity (refer to Equation (6)), T_n is the transformed temperature, T₀ (=298 K) is the initial temperature, F_C (vector) is the Coriolis force term (refer to Equation (7)), and its effect on the UHIC for the valley town can be ignored in this study [32]. F_n (vector) is the term introduced by the vertical coordinate transformation (refer to Equation (8)). k_eff is the sum of molecular and turbulent thermal conductivity, Γ is the temperature lapse rate (refer to Equation (9)), and k_n is the transformed turbulent kinetic energy (TKE). G_kn and G_bn are the TKE generation terms due to the mean velocity gradients and buoyancy, respectively. S_kn and S_εn are the terms introduced by the vertical coordinate transformation for the TKE equation (refer to Equation (10)) and the TKE dissipation rate equation (refer to Equation (11)), respectively. ε_n is the transformed TKE dissipation rate, C_1ε = 1.44, C_2ε = 1.92, C_3ε = 0.09, σ_k = 1.0, σ_ε = 1.3, C_μ = 0.09, Pr_t = 0.85. The initial values of the TKE and TKE dissipation rate are set to 10⁻⁶ m²/s² and 10⁻⁹ m²/s³, respectively, which are the same as the numerical simulation conducted by Wang and Li [33].

$${\mu }_{\mathrm{eff}}=\mu J_1+{\mu }_t$$

(6)

$${\mathit{F}}_{\mathit{C}}=\left[\begin{array}{c}{\rho }_{0}fv-{\rho }_{0}l{w}_{\mathrm{n}}J\\ -{\rho }_{0}fu\\ {\rho }_{0}luJ\end{array}\right]$$

(7)

$${\mathit{F}}_{\mathbf{n}}=\left[\begin{array}{c}0\\ 0\\ \left(J^2-1\right)\left({\rho }_0\beta \left(T_{\mathrm{n}}-T_0\right)g+{\mathit{F}}_{\mathit{C}}\right)+\xi {\rho }_0{w_{\mathrm{n}}}^{2}J-p_{\mathrm{n}}\xi \left(J+1+\xi z_{\mathrm{n}}\right)\end{array}\right]$$

(8)

$$\mathsf{\Gamma }\cong {\mathsf{\Gamma }}_{\mathrm{d}}-{\partial T_{\mathrm{p}}/\partial z}_{\mathrm{n}}$$

(9)

$$S_{k\mathrm{n}}=\beta g{\displaystyle \frac{{\mu }_t}{P{r}_t}}\left(\mathsf{\Gamma }-{\displaystyle \frac{g}{C_p}}\right)$$

(10)

$$S_{\epsilon \mathrm{n}}=C_{1\epsilon }{C}_{3\epsilon }{\displaystyle \frac{{\epsilon }_{\mathrm{n}}}{k_{\mathrm{n}}}}\beta g{\displaystyle \frac{{\mu }_t}{P{r}_t}}\left(\mathsf{\Gamma }-{\displaystyle \frac{g}{C_p}}\right)$$

(11)

where μ is the dynamic viscosity, J₁ (≈1) is the viscosity ratio, μ_t is the turbulent viscosity, J is the vertical transformation coefficient, ξ is the vertical parameter, and g/C_p is the dry adiabatic lapse rate. For detailed descriptions of the vertical transformation coefficient and vertical parameter, refer to the study by Wang and Li [33].

User-defined functions of the commercial software ANSYS FLUENT 2020 R1 are used to implement the CSCFD model application. The standard k-ε model is employed to solve the turbulence equations [34], and the standard wall function is utilized to handle near-wall regions. The Boussinesq approximation is applied to deal with the change in density due to the change in temperature [33]. The PISO algorithm and PRESTO! schemes are used for pressure–velocity coupling and pressure discretization, respectively. In the transient calculations, the time step size is set to 1 s. During the iterative calculation at each time step, the convergence is considered to have been achieved when the iterative residuals reach 10⁻⁴ for the mass, velocity, and turbulence parameters [35], as well as 10⁻⁶ for the energy parameter [36].

2.3. Grid Sensitivity Analysis

ICEM CFD is utilized for computational mesh generation, and quadrilateral grid cells are generated, as displayed in Figure 3a. For the x-direction, the grid distribution for both the valley floor and slope surface is uniform and the grid spacing is L_m, and the grid spacing of the top surface of Loess Tableland gradually increases from the slope top to the lateral boundary, and the minimum and maximum grid spacings of the top surface are L_m and 2 L_m, respectively, with a grid spacing ratio of 1.1. For the z-direction, the closer to the solid surface is, the smaller the grid spacing becomes, where the solid surfaces include the valley floor, slope surface and top surface of Loess Tableland. The minimum grid spacing near the solid surfaces is 0.1 m to ensure that the target y⁺ value is greater than 30. The maximum grid spacing near the top of the computational domain is L_m and the grid spacing ratio is also 1.1. Four grid spacings are selected for Case 1, namely the coarse grid spacing (L_m = 20 m), basic grid spacing (L_m = 12 m), fine grid spacing (L_m = 8 m) and finer grid spacing (L_m = 2.5 m), and these grid numbers are 1.18 × 10⁴ (coarse grid), 2.35 × 10⁴ (basic grid), 4.67 × 10⁴ (fine grid) and 3.90 × 10⁵ (finer grid), respectively. The simulation results with the four grid numbers are shown in Figure 3b,c, which demonstrate that the fine grid can be used for the numerical simulation of Case 1, joint numerical accuracy and computational speed. Therefore, the principle of grid generation for the other cases follows that of the fine grid for Case 1.

2.4. Model Validation

In order to validate the accuracy of the CSCFD model in simulating the UHIC for the valley town, the evolution of the UHIC under calm and stable weather is simulated by the water tank experiment [37]. Based on the Π theorem, a real Loess Tableland valley town can be scaled down to a physical model in the water tank, and the sketch of the water tank experiment is illustrated in Figure 4. Two peristaltic pumps control the salt water and pure water flow rates, respectively, and the salt water flow rate decreases gradually with time, while the pure water flow rate increases gradually with time. After mixing well in the mixer, salt water and fresh water are injected into the water tank, and thus the water body with a decreasing density from bottom to top is formed (stable stratification). The density lapse rate created is equal to −26 kg/m⁴. The inside dimensions of the water tank are 1.52 m × 1.12 m × 0.8 m, and the dimensions of the copper plate placed in the center of the water tank are 0.2 m × 1 m × 0.03 m. The copper plate has an electric heating function to simulate the town area, and the urban heat flux is set to 4742.52 W/m². The town area is situated between two Loess Tablelands, and the area between the Loess Tableland and the urban area is a suburb of 0.04 m in length. Both the suburb and the Loess Tableland are made of acrylic through 3D printing. The water tank is constructed with 12 mm-thick ultra-clear toughened glass, which offers excellent light transmittance. This characteristic is particularly advantageous for velocity field measurements using a particle image velocimetry (PIV) system. The PIV system (Dantec Dynamics A/S) mainly consists of the double-pulse YAG laser, sCMOS camera, timer box, Dynamic Studio software 7.6 and workstation. The seeding particles are PSP-20 μm and PSP-50 μm, and the mixing of the two particles allows for a sufficient number of seeding particles to be present in different density layers. For each set of experimental data, the experimental images are acquired at a frequency of 10 Hz and recorded continuously for 60 s, and these 600 images can be used to obtain the proper mean convective speed by means of the adaptive PIV algorithm of the Dynamic Studio commercial software [38].

The physical model of the CSCFD validation case is the same as that of the water tank experiment, and the CSCFD model is set up in Section 2.2. At the 8th minute of town heating, a city-scale closed circulation has formed over the valley floor, and thus the mean flow field at the 8th min is used to analyze. The mean velocity fields of CSCFD model results and water tank results are analyzed qualitatively, as shown in Figure 5a. The CSCFD model accurately simulates the typical characteristics of city-scale closed circulation, including the lower convergent inflow, upward airflow, upper convergent outflow, and internal shear layer between the inflow and the outflow. It should be noted that the lower convergent inflow on the right side and the upward airflow are weak, because the heat flux at the copper plate surface is not homogeneous and not constant, which are two effects that cannot be avoided in the experiment. The vertical distributions of horizontal wind speed at the urban edge from both CSCFD model results and water tank results are analyzed quantitatively, as depicted in Figure 5b. The vertical distribution for the CSCFD model agrees with that for the water tank, and the CSCFD model accurately predicts the shape of the vertical distribution for the water tank. Additionally, Figure 5c shows that the percentage errors of the two vertical distributions are mainly within 20%. Therefore, the results of the CSCFD model simulating the UHIC for the valley town are acceptable.

3. Results and Discussion

The study investigates the evolution characteristics of the UHIC over negative terrain. Firstly, two flow patterns of the UHIC are analyzed. Secondly, two important parameters of the second flow pattern in the transient stage are obtained, which are the start time and the outflow time, and the sensitivity analysis of the influencing factors is conducted. Finally, two key parameters of the second flow pattern in the quasi-steady stage, the mixing height and the heat island intensity, are obtained and analyzed. The sensitivity analysis adopts the range analysis of orthogonal experiments, and three steps are needed to complete the range analysis: the first step is to select the influencing factors and their levels; the second step is to calculate the range value of each influencing factor by comparing the experimental results under different levels; and the third step is to rank the influencing factors in descending order of the range values.

3.1. Two Airflow Patterns for Urban Heat Island Circulation

Available studies have shown that the UHIC for the flat city can undergo two stages when it evolves to the quasi-steady state. The first stage is the transient stage, whose flow pattern is the thermal convection, and the second stage is the quasi-steady stage, whose flow pattern is the city-scale closed circulation [39]. Similarly, the evolution process of the UHIC for the valley town can also be divided into two stages.

In the transient stage, the airflow pattern of the UHIC for the valley town is the thermal convection and city-scale closed circulation. The thermal convection is shown in (i) of Figure 6a,b, where the convective rolls move towards the urban center under the effect of the pressure difference between the town area and the suburban area [40,41]. This phenomenon is similar to the thermal convection for the flat city during the transient stage [39]. When the convective rolls merge in the urban center to form a city-scale closed circulation, the mixing height of the city-scale closed circulation is low due to the effect of the stable stratification. Significantly, the city-scale closed circulation can be obstructed by Loess Tablelands in the horizontal direction, which causes that the upper divergent outflow and lower convergent inflow cannot be sufficiently exchanged for heat with the background air, and eventually the heat of the closed circulation gradually accumulates in the valley space, meanwhile the airflow temperature in the valley space gradually increasing. This will cause the stable stratification on the top of the closed circulation to be gradually destroyed and the closed circulation continues to expand in the vertical direction, as shown in (ii) of Figure 6a,b. Therefore, the city-scale closed circulation in the transient stage can continue to develop.

When the city-scale closed circulation expands to a certain height, the upper divergent outflow and lower convergent inflow have sufficiently affected the space on the top surface of the Loess Tableland, and at this time, the circulation and background air can sufficiently exchange heat, and thus the heat in the valley space no longer accumulates. Eventually, the city-scale closed circulation no longer expands in the vertical direction, indicating that it reaches the quasi-steady stage, as seen in (iii) of Figure 6a,b. In the quasi-steady stage, the airflow pattern of the UHIC for the valley town is similar to that for the flat city, both of which are the city-scale closed circulations [42]. There is a curved internal shear layer between the upper divergent outflow and the lower convergent inflow in the quasi-steady stage, which is caused by the fact that the upper divergent outflow is in a non-equilibrium state under the alternating forcing of the buoyancy and inertia force [43].

The evolution of the temperature field for different cases is displayed in Figure 7, which shows that the airflow temperature in the valley space is getting higher and higher during the transient stage, and the airflow temperature in the space on the top surface of the Loess Tableland is also obviously affected during the quasi-steady stage. Note that the vertical distribution of background air temperature for Case 6 is different from that for Case 22, and the atmospheric stratification for Case 22 is very stable and referred to as the inversion layer. The water tank results described in Section 2.4 also qualitatively validate the above conclusions well, and the flow fields at different moments captured by the camera in the PIV system are shown in Figure 8. A large number of white dots in the figure are polyamid seeding particles (PSP), which are used to trace the flow. The reason for the presence of bright streaks in the figure could be the significant change in the refractive index due to the large increase in fluid temperature, which in turn affects the camera shooting effect. In order to prove the above reason, a water tank experiment of the UHIC evolution for the flat city is also conducted, and it is found that there are no bright streaks during the experiment. This is due to the fact that convective heat transfer between the heated fluid and the ambient fluid is more adequate on the flat terrain, and the fluid temperature is not increased significantly. Therefore, the bright streaks can reflect the change in fluid temperature to some extent. Firstly, the moment shown in Figure 8i is shortly after the town heating, and the increase in fluid temperature is not significant, thus no bright streaks appear over the town area. Secondly, at the moment shown in Figure 8ii, the city-scale closed circulation is in the transient stage, and the bright streaks appear due to the blocking effect of the Loess Tablelands on both sides of the town area, resulting in the accumulation of heat and the increase in fluid temperature in the valley space. And finally, at the moment shown in Figure 8iii, the city-scale closed circulation has evolved to the quasi-steady state, and it has affected the area on the top surface of the Loess Tableland. At this moment in time, convective heat exchange between the city-scale closed circulation and the ambient fluid is sufficiently adequate, and the circulation temperature is no longer increased, while the development of bright streaks reaches a quasi-steady state.

3.2. Start Time and Outflow Time of City-Scale Closed Circulation

3.2.1. Start Time of City-Scale Closed Circulation

During the transient stage, air pollutants released from the town area can only be transported by the thermal convection at the local scale, and thus, it is very easy to cause local air pollution. When the city-scale closed circulation is formed, the airflow circulates at the city scale, which is conducive to the transport of air pollutants. Therefore, the formation of city-scale closed circulation represents a shift in the airflow pattern over the town area and an increase in the transport scale of air pollutants. The dimensionless start time of city-scale closed circulation for the flat city is related to the Froude number (Fr) by t_csN = 2.174/Fr [29]. The dimensionless start times for the valley town and flat city are plotted in Figure 9, and it can be seen that the dimensionless start time of the former is earlier than that of the latter. Dimensionless start time of the former decreases by 10.47% on average compared to that of the latter, and the maximum value of the decrease can reach 31.19%. Therefore, the Loess Tablelands on both sides of the town area can promote the convective rolls moving toward the urban center.

The prediction of the start time for the valley town has important practical applications, as well as providing the theoretical basis for conducting water tank experiments with relevant contents. When the city-scale closed circulation has just formed, the heating time at this moment is the start time of the city-scale closed circulation. As described in Section 2.1, there are seven core variables that influence the UHIC for the valley town. When the heating time is long enough and continuous, a case that includes the six variables shown in Table 2 can determine a start time of the city-scale closed circulation. Therefore, based on the multiple linear regression method, the start time is predicted by applying these six variables as follows:

$$t_{\mathrm{cs}}={\displaystyle \frac{1}{N}}\left(1.60D/H+0.46L/H+4.42W/H-18.88g\beta Q_{\mathrm{i}}/H^2{N}^3\right)$$

(12)

where t_cs is the start time in seconds. Four performance metrics of the predictive model (Equation (12)) are shown in

Table 3. Performance metrics of the predictive model.

Predictive Model	RSS	RMSE	R2	VIF
Equation (12)	1294.44	7.19	0.85	6.67
Equation (13)	7.79	0.56	0.97	33.33

. The R² of 0.85 indicates that the fitting result is good, and the VIF of 6.67 indicates that there is a multicollinearity problem, but it is weak. According to the sensitivity analysis of different factors, as shown by the range (R_s) values in

Table 4. The sensitivity analysis.

Index	Range (Rs)						In Order of Priority
	D	Qu	∂Tp/∂z	L	H	θ
tcs	432.00	828.00	204.00	132.00	156.00	84.00	Qu > D > ∂Tp/∂z > H > L > θ
tos	4038.60	5500.40	3238.40	3678.00	6970.20	3548.60	H > Qu > D > L > θ > ∂Tp/∂z
zic	42.75	146.95	215.33	21.02	221.26	22.96	H > ∂Tp/∂z > Qu > D > θ > L
∆Tqs	0.47	1.86	2.16	0.53	2.02	0.52	∂Tp/∂z > H > Qu > L > θ > D

, the factor that influences the start time the most is the urban heat flux, followed by the urban length, potential temperature lapse rate, slope height, suburban length, and the factor that influences the start time the least is the slope angle. Figure 10 shows the trends of the start time with the urban heat flux and urban length, which reveal that the start time is negatively correlated with the urban heat flux (Figure 10a), meaning that the larger the urban heat flux, the earlier the dimensionless start time. At the same time, the start time is positively correlated with the urban length (Figure 10b), meaning that the greater the urban length, the later the dimensionless start time.

3.2.2. Outflow Time of City-Scale Closed Circulation

Due to the blocking effect of the Loess Tablelands on both sides of the town area, the mixing height of the city-scale closed circulation will continue to increase during the transient stage. When the mixing height increases to a certain height, the upper divergent outflow of the city-scale closed circulation can affect the space on the top surface of the Loess Tableland, which implies that the airflow can be exchanged between the valley space and the space on the top surface of the Loess Tableland. As illustrated by the streamtraces in Figure 11, the city-scale closed circulation connects these two spaces. Therefore, the scenario in which a city-scale closed circulation develops to the top surface of Loess Tableland represents that air pollutants or heat accumulated in the valley space by the UHIC can be transported to the space on the top surface of Loess Tableland, which is conducive to improve the air quality and mitigate the urban heat island in the valley space. The heating time at which the upper divergent outflow begins to affect the space on the top surface of Loess Tableland is referred to as the outflow time of city-scale closed circulation, which is determined based on the fact that the maximum horizontal wind speed (u_max) is just greater than 0.1 m/s at this moment, and the maximum wind speed is always greater than 0.1 m/s (the average value of wind scale 0) for the time after this moment, as shown in Figure 11.

The sensitivity analysis of the factors influencing the outflow time of the city-scale closed circulation is conducted. According to the R_s in Table 4, the slope height has the greatest influence on the outflow time, followed by the urban heat flux, urban length, suburban length, slope angle, and the potential temperature lapse rate has the least influence on the outflow time. Figure 12 shows the trends of the outflow time with the slope height and urban heat flux, which indicate that the outflow time is positively correlated with the slope height (Figure 12a), meaning that the higher the slope height, the later the outflow time. At the same time, the outflow time is negatively correlated with the urban heat flux (Figure 12b), meaning that the larger the urban heat flux, the earlier the outflow time. Therefore, from the perspective of the outflow time, the town area should be built on the valley floor where the slope heights are small. This suggestion will shorten the outflow time, and thus air pollutants emitted from the town area can be transported to the top surface areas of the Loess Tableland earlier, which is conducive to improving the air quality in the town area.

3.3. Mixing Height and Heat Island Intensity in Quasi-Steady Stage

3.3.1. Mixing Height in Quasi-Steady Stage

Under calm and stable weather, the background wind speed is very weak, and thus, the mixing height is crucial to determine the atmospheric environment capacity of the city. The mixing height is defined as the height at which the maximum negative difference of temperature occurs [8], and it denotes the maximum height that can be reached by airflow motion and air pollutant transport. The trends of mixing height with time for Case 8 and Case 17 are shown in Figure 13, and the trends of mixing height for the other cases are similar to those for Case 7 and Case 8. After the formation of the city-scale closed circulation, the mixing heights continue to increase for both cases. This phenomenon is caused by the negative terrain of the Loess Tableland valley town. The mixing height for Case 8 reaches the quasi-steady stage after about 2 h of heating time, while the mixing height for Case 17 reaches the quasi-steady stage after about 4 h of heating time.

The dimensionless mixing height in the quasi-steady stage for the flat city is a function of the Fr, and the result of the water tank experiment is z_ic/D = 2.86 Fr [8]. The trend of the dimensionless mixing height with the Fr for the valley town is shown in Figure 14, and the values of the dimensionless mixing height are the average values in the quasi-steady stage. There are two differences between the trends for the valley town and the flat city. Firstly, the dimensionless mixing height for the valley town is significantly larger than that for the flat city. Compared with the latter dimensionless mixing heights, the former dimensionless mixing heights increase by 56.80% on average, and the maximum value of the increase can reach 117.20%. Secondly, the negative terrain of the Loess Tableland valley town causes the Fr to no longer be the most important similarity parameter for the UHIC [8]. The dimensionless mixing height for the valley town cannot be expressed by Fr alone. The seven core variables described in Section 2.1 may predict the mixing height. However, the mixing height in the quasi-steady stage is not dependent on the heating time, and therefore, the mixing height in the quasi-steady stage can be determined by the six variables shown in Table 2. Based on the multiple linear regression method, these six variables are applied to predict the mixing height in the quasi-steady stage as follows:

$$z_{\mathrm{ic}}=H\left(0.11D/H+0.33L/H+0.66W/H+3.17g\beta Q_{\mathrm{i}}/H^2{N}^3\right)$$

(13)

where z_ic is the mixing height in meters. Four performance metrics for Equation (13) are shown in Table 3. The fitting result is very good, but the multicollinearity problem is significant. Future research will solve the multicollinearity problem through the Principal Component Analysis, Ridge Regression Algorithm or Lasso Regression Algorithm, while adding urban canopy height to the set of independent variables. The atmospheric environment capacity can be determined based on the mixing height, and then related departments according to the atmospheric environment capacity can provide control strategies for the emission of urban air pollutants.

According to the R_s in Table 4, the slope height has the greatest influence on the mixing height in the quasi-steady stage, followed by the potential temperature lapse rate, urban heat flux, urban length, slope angle, and suburban length has the least influence on the mixing height. Figure 15 shows the trends of the mixing height with the slope height and potential temperature lapse rate, which indicate that the mixing height is obviously positively correlated with the slope height (Figure 15a), meaning that the higher the slope height, the greater the mixing height. At the same time, the mixing height is obviously negatively correlated with the potential temperature lapse rate (Figure 15b), meaning that the larger the potential temperature lapse rate, the smaller the mixing height.

3.3.2. Heat Island Intensity in Quasi-Steady Stage

The heat island intensity in the quasi-steady stage plays an important role in urban thermal environment assessment [44]. The heat island intensity for the flat city is defined as the near-surface air temperature difference between the urban center and the surrounding surface [8]. In this study, the heat island intensity for the valley town is determined by the difference between the average temperature at a height of 1.5 m above the urban surface and the initial temperature. The trends of heat island intensity with time are shown in Figure 13, and since the heat island intensity trends are similar for all the cases, only two of them are shown (Case 8 and Case 17). After the formation of the city-scale closed circulation, the trends of the heat island intensity are very similar to those of the mixing height, which continue to increase and then reach the quasi-steady state. The heating time needed to reach the quasi-steady state for the heat island intensity of Case 8 and Case 17 is about 2 h and 4 h, respectively. The trends of heat island intensity and mixing height with time are caused by the fact that the city-scale closed circulation has two stages, the transient stage and the quasi-steady stage, which are present due to the blockage effect of the Loess Tablelands. For the variables of the Loess Tableland valley town counted in the present study (Table 2), the city-scale closed circulation for all cases has these two stages. However, for other scenarios, such as high slope heights, small slope angles and small urban heat fluxes, the quasi-steady stage of city-scale closed circulation may not exist, which is because the city-scale closed circulation cannot develop to the top surface areas of the Loess Tableland, and thus cannot reach the quasi-steady state. Future research will focus on some special scenarios.

The heat island intensity in the quasi-steady stage for the flat city is a function of u_DN/gβ, and the result of the water tank experiment is ΔT_qs = 1.16 u_DN/gβ [8]. The trend of the heat island intensity with u_DN/gβ for the valley town is shown in Figure 16, and the values of the heat island intensity are the average values in the quasi-steady stage. Compared with the heat island intensity for the flat city, the heat island intensity for the valley town increases by 128.68% on average, and the maximum value of the increase can reach 221.28%. The heat island intensity is significantly greater for the valley town than for the flat city, implying that the residents in the valley town are confronted with higher heat stress, and their demand for space cooling is larger [45].

The sensitivity analysis of the factors influencing the heat island intensity in the quasi-steady stage is conducted. According to the R_s in Table 4, the factor that influences the heat island intensity the most is the potential temperature lapse rate, followed by the slope height, urban heat flux, suburban length, slope angle, and the factor that influences the heat island intensity the least is the urban length. Figure 17 shows the trends of the heat island intensity with the potential temperature lapse rate and slope height, which reveal that the heat island intensity is positively correlated with both the potential temperature lapse rate and the slope height (Figure 17a,b), meaning that the larger the potential temperature lapse rate, or the higher the slope height, the stronger the heat island intensity. Therefore, for the Loess Tableland valley town where temperature inversion is frequent, the town area should take some methods to decrease urban heat flux, such as reducing energy consumption and increasing vegetation coverage [46,47], in order to mitigate urban heat island intensity. In addition, the location of the town area selected at the valley floor, where the slope heights are small, is also a useful measure that can mitigate urban heat island intensity.

4. Limitation

It is necessary to acknowledge two major limitations in this study. First, in order to focus on the effect of the negative terrain of the Loess Tableland valley town on the evolution characteristics of the UHIC, only urban heat flux is considered in the town area, and the building complex is not considered. The building complex can increase the complexity of the UHIC characteristics [48]. The study by Saitoh and Yamada [49] shows that the presence of a building complex can cause a single UHIC to turn into multiple UHICs. The results of Wang and Li [33] suggest that the height of the urban canopy layer slightly influences the potential temperature in the center and influences the entrainment strength of the urban plume. Second, the potential temperature lapse rate in this paper is set to be a constant, whereas it is usually not a constant in the real atmosphere. It has been demonstrated that the change in potential temperature lapse rate with height has a significant influence on the airflow pattern of the UHIC and the transport process of air pollutants [38,50]. Based on this study on the effect of the negative terrain alone on the evolution characteristics, future research will extend to the effect of building complex and atmospheric stability on the evolution characteristics, and, subject to favorable conditions, field measurements will be conducted to validate the numerical results.

5. Conclusions

In order to improve the wind and thermal environments of the Loess Tableland valley town under calm and stable weather, the CSCFD model is used to explore the evolution characteristics of the UHIC. With the analysis of velocity and temperature fields of the UHIC evolution, the start time and outflow time of the city-scale closed circulation, as well as the mixing height and heat island intensity in the quasi-steady stage, the primary difference between the UHIC for the valley town and that for the flat city is that the former city-scale closed circulation can expand vertically during the transient stage. The salient findings of the study are summarized below:

(1) For the valley town, there are two flow patterns of UHIC in the transient stage, which are the thermal convection and the city-scale closed circulation. After the formation of the city-scale closed circulation, the city-scale closed circulation will continue to expand in the vertical direction. When the city-scale closed circulation exceeds a certain height above the slope height, it will stop expanding vertically and eventually reach the quasi-steady stage. Meanwhile, the airflow temperature in the valley space is increasing during the transient stage and no longer increasing with time during the quasi-steady stage.

(2) The dimensionless start time of the city-scale closed circulation for the valley town is reduced by 10.47% compared to that for the flat city, and the start time is predicted by the multiple linear regression. According to the magnitude of influence, the order of six factors influencing the start time is urban heat flux > urban length > potential temperature lapse rate > slope height > suburban length > slope angle, and the order of six factors influencing the outflow time is slope height > urban heat flux > urban length > suburban length > slope angle > potential temperature lapse rate. The town area should be built on the valley floor, where the slope heights are small, to shorten the outflow time.

(3) The dimensionless mixing height and heat island intensity in the quasi-steady stage for the valley town increase by 56.80% and 128.68%, respectively, compared with those for the flat city, indicating that the urban heat island effect in the valley town is more severe. According to the magnitude of influence, the order of six factors influencing the mixing height is slope height > potential temperature lapse rate > urban heat flux > urban length > slope angle > suburban length, and the order of six factors influencing the heat island intensity is potential temperature lapse rate > slope height > urban heat flux > suburban length > slope angle > urban length. For the Loess Tableland valley town, where temperature inversion is frequent, the town area should take some methods that can decrease urban heat flux to mitigate urban heat island intensity.