A Linear Theory of Local Thermal Circulations

Abstract

The dynamics of local thermal circulations (LTCs) are examined by constructing a linear theory based on Boussinesq equations in the planetary boundary layer (PBL). Linear theory arranges LTCs into a sixth-order partial differential equation of temperature, which can be solved by using the Fourier transform and inverse Fourier transform. The analytic solution suggests that the horizontal distribution of the temperature anomaly is basically determined by surface heating, while the vertical distribution of the temperature anomaly is a combination of exponential decay and an Ekman spiral. For shallow PBL cases where the Ekman elevation is much smaller than the vertical scale of motion, the higher-order partial differential terms that represent the Ekman spiral structure can be ignored so that the equations reduce to a second-order partial differential equation. Compared with the numerical results, this so-called low-order approximation does not induce dramatic errors in the temperature distribution. However, to avoid distortion in the forced atmospheric circulation, the eddy viscosity in the motion equations should be replaced with the Rayleigh form, which is the common practice in LTCs. For deep PBL cases where the Ekman elevation is comparable to the vertical scale of motion, both the exponential decay and Ekman spiral structure play roles in the forced atmospheric circulation. The most significant influence is that there exist three additional compensating forced circulation cells that surround the direct forced circulation cell.

1. Introduction

Local thermal circulations (LTCs), mainly driven by surface differential heating, often take the form of a breeze-like circulation with a near-surface air mass blowing in the planetary boundary layer (PBL) from a cold to warm surface and an upper-level compensating return flow above the PBL [1]. They have different names in different types of underlying surfaces. For example, a sea-land breeze has long been a term used to describe the common phenomenon caused by temperature differences between land and sea [2]. Similarly, a lake–land breeze is caused by a temperature contrast between land and lake [3]. In urban regions, urban breeze circulation (UBC), or urban heat island circulation, characterizes the local wind-driven thermal difference between urban and rural areas [4]. In addition, an oasis breeze circulation and a mountain–valley breeze circulation are both linked with the difference in thermal heating due to heterogeneous surface properties.

LTCs are generally studied through commonly applied Boussinesq equations in PBLs. Theoretical studies have discussed the dynamics of LTCs by analytically solving the equations under different simplifications and hypotheses [5]. For example, Rotunno [6] discussed the influence of Coriolis parameter and buoyance frequency on the atmospheric response. Dalu and Pielke [7] presented theoretical results concerning sea breeze intensity and its inland penetration as a function of latitude and friction. Qian et al. [8] modeled equatorial coastal circulation in terms of the linear wave response to a diurnally oscillating heat source gradient in a background wind. Drobinski et al. [9] noted that the sea breeze circulation is tilted toward slanted isentropes associated with thermal wind. Jiang [10] assessed complex coastline shapes and geometries, the Earth’s rotation, and background wind effects by using a three-dimensional linear model that is solved by fast Fourier transform. Seo et al. [11] studied the dynamics of a reversed UBC in the context of atmospheric response to specified thermal forcing. Li and Chao [12,13] emphasized the specific thermal forcing and expressed LTCs as analytic functions of the temperature distribution. Previous advances have greatly promoted the understanding of LTCs. In addition, observational studies have presented the temporal and spatial structures of LTCs (e.g., [14,15]), while numerical modeling studies have contributed to revealing circulation features and the relationships among circulation and different variables or parameters, such as diurnal propagation (e.g., [16,17]), convection (e.g., [18]), background winds and wind shear (e.g., [19,20]), and heat flux (e.g., [21]).

To analytically assess the dynamics of LTCs, one common practice is to replace the eddy viscosity coefficient with a Rayleigh-type friction coefficient to reduce the control equations to a second-order partial differential equation. A corresponding wave equation could be derived by applying the Fourier transform. The wave equation could thus be easily solved under specific boundary conditions. The solution could then be obtained by applying inverse Fourier transform. Although the Rayleigh form coefficient not only considers the effect of friction but also simplifies the mathematical derivation, its simple form is bound to lose some vertical structural characteristics featured by the eddy viscosity. Therefore, it is necessary to discuss the dynamics of LTCs in equations with eddy viscosity coefficients. Walsh [5] reduced linearized Boussinesq equations with viscosity and conduction to a group of sixth-order partial differential equations that he formally solved but without further discussion. Sun and Orlanski [22] took similar equations but solved them numerically. Sang et al. [23] introduced viscosity in a thermodynamic equation but the Rayleigh form of friction in motion equations. He then derived a fourth-order equation of temperature to assess the dynamics of UBC. Li and Chao [12,13] applied viscosity in motion equations, but they specified a complete known temperature distribution to emphasize the effect of local thermal forcing. The fact that the basic dynamics of LTCs with viscosity in both thermodynamic and motion equations have yet to be extensively studied motivates this research. Therefore, this paper comprehensively investigates the dynamics of LTCs in steady, linearized, Boussinesq equations with viscosity and conduction. The thermodynamic and motion equations and solutions are introduced in Section 2. LTC dynamics under different prerequisites are presented and discussed in Section 3. A summary ends the paper in Section 4.

2. The Linear Theory

2.1. Thermodynamic Equation

The steady thermodynamic equation without localized heating (e.g., see [5,17]) is generally written as

$$\frac{N^2\overline{T}}{g}w={\kappa }_z\frac{{\partial }^{2}T}{\partial z^2}$$

(1)

where $w$ is the vertical velocity; $N$ is the buoyancy frequency; $\overline{T}$ and $T$ are the reference temperature of the stationary atmospheric background and departure of temperature with respect to the reference temperature, respectively; $g$ is the acceleration of gravity; and ${\kappa }_z$ is the eddy coefficient of viscosity. A common practice (e.g., see [11]) is to add localized heating to the right-hand side of Equation (1) and to replace ${\kappa }_z\frac{{\partial }^{2}T}{\partial z^2}$ with $-\alpha T$ for simplicity, where $\alpha$ is called the Newtonian cooling coefficient. However, as mentioned above, the conduction term is not simplified in this paper, and localized heating is not introduced in the thermodynamic equation but in the lower boundary condition.

Equation (1) does not consider the influence of radiative transfer of heat in the atmosphere. Kuo [24] divided the absorption spectrum of terrestrial radiation into strongly and weakly absorbed regions and introduced two mean absorption coefficients for these two groups of regions. He noted that the influence of the strongly absorbed regions is equivalent to added diffusion, while that of the weakly absorbed regions is Newtonian cooling. He applied this to the problem of the thermal interaction between the atmosphere and the underlying Earth and found that the temperature changes within the first few hundred meters from the Earth’s surface could be predicted accurately by the model. Inspired by the global energy balance, Chao and Chen [25] identified the integral constant unsolved by Kuo [24]. Based on their investigation, the thermodynamic equation is modified to

$$\left(N^2\frac{\overline{T}}{g}\right)w=\left({\kappa }_z+{\kappa }_r\right)\frac{{\partial }^{2}T}{\partial z^2}-\tau T$$

(2)

where ${\kappa }_r$ can be called the equivalent radiative eddy coefficients of viscosity, implying that the strongly absorbed regions play roles in increasing the eddy coefficient of conduction, and $\tau$ is called the Newtonian cooling coefficient, reflecting that the weakly absorbed regions have a damping effect. According to Li et al. [26], ${\kappa }_z$ is much larger than ${\kappa }_r$. Therefore, it is reasonable to set $\kappa ={\kappa }_z+{\kappa }_r\approx {\kappa }_z$. Because the thermal perturbation is mainly predominant at the lower level, the corresponding perturbation temperature should vanish when $z$ is sufficiently high, namely

$$z\to \infty ,T\to 0$$

(3)

The lower boundary condition is generally written as

$$-{\kappa }_z\frac{\partial T}{\partial z}=Q\left(x\right)\equiv Q_{0}q\left(x\right)$$

(4)

where $Q$ is the perturbation thermal flux distribution at the lower boundary, $Q_0=\frac{\kappa T_0}{z_0}$ is its magnitude, $T_0$, $z_0$ are the characteristic scale of the temperature perturbation and vertical motion, and $q\left(x\right)$ is its nondimensional distribution. Following Rotunno [6], the lower boundary heating is set to

$$q\left(x\right)=\frac{2}{\pi }{\tan }^{-1}\left(\frac{x}{\gamma }\right)$$

(5)

to represent the two types of heterogeneous underlying heating. $\gamma$ is a parameter to control how steep the transition is in moving from one type of heating to another. When $\gamma \to 0$, $q\left(x\right)$ is a step function that is discontinuous at $x=0$. The heating function is also similar to the one applied by Drobinski et al. [9].

2.2. Motion Equation

Steady, hydrostatic, and linearized Boussinesq equations with no background wind can be written as

$$-fv=-\frac{1}{{\rho }_0}\frac{\partial p}{\partial x}+K\frac{{\partial }^{2}u}{\partial z^2}$$

(6)

$$fu=K\frac{{\partial }^{2}v}{\partial z^2}$$

(7)

$$\frac{1}{{\rho }_0}\frac{\partial p}{\partial z}=\frac{g}{\overline{T}}T$$

(8)

$$\frac{\partial u}{\partial x}+\frac{\partial w}{\partial z}=0$$

(9)

where $f$ is the Coriolis parameter; $K$ is the eddy coefficient of viscosity for motion and is set to be equal to $\kappa$; ${\rho }_0$ and $\overline{T}$ are the reference density and temperature of the stationary atmospheric background, respectively; and $p,\left(u,v,w\right)$ represent the perturbations of the pressure and velocity components in the $\left(x,y,z\right)$ directions with respect to the stationary atmospheric background, respectively. According to Equation (9), stream function can be introduced to describe the circulation. The boundary conditions are taken as

$$z=0,w=0$$

(10)

$$z=0,\frac{\partial }{\partial z}\left(K\frac{{\partial }^{2}u}{\partial z^2}\right)=\frac{\partial }{\partial z}\left(K\frac{{\partial }^{2}v}{\partial z^2}\right)=\frac{\partial }{\partial z}\left(K\frac{{\partial }^{2}w}{\partial z^2}\right)=\frac{\partial }{\partial z}\left(\kappa \frac{{\partial }^{2}T}{\partial z^2}\right)=0$$

(11)

The boundary condition Equation (10) means a flat lower boundary where the vertical velocity vanishes at the ground surface, while the boundary condition Equation (11) suggests a constant flux at the surface due to strong turbulent mixing. Both boundary conditions are commonly applied in theoretical analysis.

2.3. Solution

According to Equations (6)–(9), a relationship between $w$ and $T$ can be easily derived as follows:

$$h_E^4\frac{{\partial }^{4}w}{\partial z^4}+w=-\frac{gK}{f^2\overline{T}}\frac{{\partial }^{2}T}{\partial x^2}$$

(12)

where $h_E=\sqrt{K/f}$ is the Ekman elevation. A sixth-order partial differential equation of variable $T$ can be derived by substituting Equation (12) into Equation (2) to eliminate $w$. Thus,

$$h_E^4\kappa \frac{{\partial }^{6}T}{\partial z^6}-h_E^4\tau \frac{{\partial }^{4}T}{\partial z^4}+\kappa \frac{{\partial }^{2}T}{\partial z^2}+K{\left(\frac{N}{f}\right)}^2\frac{{\partial }^{2}T}{\partial x^2}-\tau T=0$$

(13)

By introducing the nondimensional variables $\tilde{z}=z/z_0$, $\tilde{x}=x/L$, and $\tilde{T}=T/T_0$, the nondimensional form of Equation (13) could be written as (the mark ‘~’ has been omitted)

$$A_E^4\frac{{\partial }^{6}T}{\partial z^6}-A_{\kappa }^2{A}_E^4\frac{{\partial }^{4}T}{\partial z^4}+\frac{{\partial }^{2}T}{\partial z^2}+{\mathrm{Bu}}^2\frac{{\partial }^{2}T}{\partial x^2}-A_{\kappa }^{2}T=0$$

(14)

where

$$A_E=\frac{h_E}{z_0},A_{\kappa }^2=\frac{\tau z_0^2}{\kappa },{\mathrm{Bu}}^2={\left(\frac{z_{0}N}{Lf}\right)}^2$$

(15)

are three nondimensional parameters. $A_E$ is the ratio between the Ekman elevation and vertical scale of motion. This indicates the relative importance between the characteristic PBL depth and vertical motion depth. If $A_E$ is less (larger) than unity, the characteristic PBL depth is smaller (larger) than the corresponding vertical motion depth. If yes (no), we could say that the PBL is shallow (deep) when compared with the vertical scale of motion. $A_{\kappa }^2$ represents the relative importance between Newtonian cooling and thermal conduction. A larger (smaller) $A_{\kappa }^2$ suggests that Newtonian cooling plays a major (minor) role in the thermodynamic Equation (2). ${\mathrm{Bu}}^2$ is the Burger number in the PBL. It is the ratio square between the Rossby deformation radius and horizontal motion scale.

By applying the Fourier transform in the horizontal dimension

$$T_m\left(z\right)\equiv {\displaystyle {\int }_{-\infty }^{+\infty }T\left(x,z\right)e^{-imx}dx}=F\left[T\left(x,z\right)\right]$$

(16)

Equation (14) becomes

$$A_E^4\frac{{\partial }^6{T}_m}{\partial z^6}-A_{\kappa }^2{A}_E^4\frac{{\partial }^4{T}_m}{\partial z^4}+\frac{{\partial }^2{T}_m}{\partial z^2}-{\mathrm{Bu}}^2{m}^2{T}_m-A_{\kappa }^2{T}_m=0$$

(17)

The corresponding characteristic equation with characteristic root $\lambda$ is

$${\lambda }^6-A_{\kappa }^2{\lambda }^4+A_E^{-4}{\lambda }^2-A_E^{-4}\left({\mathrm{Bu}}^2{m}^2+A_{\kappa }^2\right)=0$$

(18)

Setting $\omega ={\lambda }^2-\frac{A_{\kappa }^2}{3}$, it can be reduced to a standard cubic equation of $\omega$

$${\omega }^3+p\omega +q=0$$

(19)

where

$$p=\left(A_E^{-4}-\frac{A_{\kappa }^4}{3}\right), q=-\left[\frac{2}{27}{A}_{\kappa }^6+A_E^{-4}\left({\mathrm{Bu}}^2{m}^2+\frac{2}{3}{A}_{\kappa }^2\right)\right]$$

(20)

It is easy to derive that the discriminant

$$\mathsf{\Delta }={\left(\frac{q}{2}\right)}^2+{\left(\frac{p}{3}\right)}^3>0$$

(21)

Therefore, Equation (19) has a real root (positive due to negative $q$) and a pair of conjugate complex roots, labeled ${\omega }_1$, ${\omega }_2$, and ${\omega }_3$. They are

$$\begin{array}{l}{\omega }_1=\sqrt[3]{-\frac{q}{2}+\sqrt{\mathsf{\Delta }}}+\sqrt[3]{-\frac{q}{2}-\sqrt{\mathsf{\Delta }}}\\ {\omega }_2=a\left(\sqrt[3]{-\frac{q}{2}+\sqrt{\mathsf{\Delta }}}+\sqrt[3]{-\frac{q}{2}-\sqrt{\mathsf{\Delta }}}\right)+a^2\left(\sqrt[3]{-\frac{q}{2}+\sqrt{\mathsf{\Delta }}}-\sqrt[3]{-\frac{q}{2}-\sqrt{\mathsf{\Delta }}}\right)i\\ {\omega }_2=a^2\left(\sqrt[3]{-\frac{q}{2}+\sqrt{\mathsf{\Delta }}}+\sqrt[3]{-\frac{q}{2}-\sqrt{\mathsf{\Delta }}}\right)+a\left(\sqrt[3]{-\frac{q}{2}+\sqrt{\mathsf{\Delta }}}-\sqrt[3]{-\frac{q}{2}-\sqrt{\mathsf{\Delta }}}\right)i\end{array}$$

(22)

where $a=\frac{-1+\sqrt{3}i}{2}$. Then, it is easy to obtain the corresponding six roots of Equation (18), and they are named $\pm {\lambda }_1$, $\pm {\lambda }_2$, and $\pm {\lambda }_3$, respectively, for convenience. According to the upper boundary condition, the temperature perturbation is limited at infinity height. This would require an exclusion of the vertically amplifying components featured by the roots that are positive or with a positive real part (labeled as ${\lambda }_1$, ${\lambda }_2$, and ${\lambda }_3$, respectively). The solution of Equation (18) eventually becomes

$$T_m=C_1{e}^{-{\lambda }_{1}z}+C_2{e}^{-{\lambda }_{2}z}+C_3{e}^{-{\lambda }_{3}z}$$

(23)

where $C_1$, $C_2$, and $C_3$ are three constants and need three equations to identify their specific values. Substituting it into the lower boundary condition Equation (4), we can obtain the first equation

$${\lambda }_1{C}_1+{\lambda }_2{C}_2+{\lambda }_2{C}_3=q_m$$

(24)

where $q_m\equiv F\left[q\left(x\right)\right]=-2i\frac{e^{-\gamma \left|m\right|}}{m}$. According to Equation (10), Equation (2) at the lower boundary can be written as

$$\left({\lambda }_1^2-A_{\kappa }^2\right)C_1+\left({\lambda }_2^2-A_{\kappa }^2\right)C_2+\left({\lambda }_3^2-A_{\kappa }^2\right)C_3=0$$

(25)

This is the second equation, and the third equation is based on the boundary condition Equation (11)

$${\lambda }_1^3{C}_1+{\lambda }_2^3{C}_2+{\lambda }_3^3{C}_3=0$$

(26)

The specific values of $C_1$, $C_2$, and $C_3$ can be obtained by solving the simultaneous Equations (24)–(26). They are

$$\begin{array}{l}{C}_1=-\frac{A_{\kappa }^2\left({\lambda }_2{\lambda }_3-{\lambda }_1^2\right)-{\lambda }_2^2{\lambda }_3^2}{\left({\lambda }_1-{\lambda }_2\right)\left({\lambda }_1-{\lambda }_3\right)\left[A_{\kappa }^2\left({\lambda }_1+{\lambda }_2+{\lambda }_3\right)+{\lambda }_1{\lambda }_2{\lambda }_3\right]}{q}_m\\ C_2=-\frac{A_{\kappa }^2\left({\lambda }_1{\lambda }_3-{\lambda }_2^2\right)-{\lambda }_1^2{\lambda }_3^2}{\left({\lambda }_2-{\lambda }_1\right)\left({\lambda }_2-{\lambda }_3\right)\left[A_{\kappa }^2\left({\lambda }_1+{\lambda }_2+{\lambda }_3\right)+{\lambda }_1{\lambda }_2{\lambda }_3\right]}{q}_m\\ C_3=-\frac{A_{\kappa }^2\left({\lambda }_1{\lambda }_2-{\lambda }_3^2\right)-{\lambda }_1^2{\lambda }_2^2}{\left({\lambda }_3-{\lambda }_1\right)\left({\lambda }_3-{\lambda }_2\right)\left[A_{\kappa }^2\left({\lambda }_1+{\lambda }_2+{\lambda }_3\right)+{\lambda }_1{\lambda }_2{\lambda }_3\right]}{q}_m\end{array}$$

(27)

The specific form of Equation (23) is then known. By applying the inversion Fourier transform, the temperature distribution is

$$T\left(x,z\right)\equiv \frac{1}{2\pi }{\displaystyle {\int }_{-\infty }^{+\infty }{T}_m\left(z\right)e^{imx}dm}=F^{-1}\left[T_m\left(z\right)\right]$$

(28)

Note that ${\lambda }_1$, ${\lambda }_2$, and ${\lambda }_3$ are all functions of $m$, which makes it difficult to analytically solve Equation (28).

3. Results

The most representative features of LTCs appear in the PBL. Therefore, the vertical scale of motion is set to $z_0={10}^3 \mathrm{m}$, rather than $z_0={10}^4 \mathrm{m}$, and the vertical scale for large-scale motion and the horizontal scale is set to $L={10}^5 \mathrm{m}$. The buoyancy frequency and Coriolis parameter are usually taken as $N=0.01 {\mathrm{s}}^{-1}$ and $f_0={10}^{-4} {\mathrm{s}}^{-1}$, respectively. The Rossby deformation radius in the PBL is $L_0\equiv \frac{N{z}_0}{f_0}={10}^5 \mathrm{m}$, which equals the horizontal motion scale. Therefore, the nondimensional parameter ${\mathrm{Bu}}^2=1$. The value of the eddy viscosity coefficient $\kappa$ generally varies from ${10}^0 {\mathrm{m}}^2{ \mathrm{s}}^{-1}$ to ${10}^2 {\mathrm{m}}^2{ \mathrm{s}}^{-1}$ and is set to $\kappa =5 {\mathrm{m}}^2{ \mathrm{s}}^{-1}$ in this paper to characterize a moderate viscosity. Applying such a viscosity value, the characteristic Ekman elevation $h_E=223.6 \mathrm{m}$. According to Rotunno [6], the value of the Newtonian cooling coefficient is not definitely specified and typically varies from 0 to ${10}^{-3} {\mathrm{s}}^{-1}$. A moderate Newtonian cooling is set to $\tau ={10}^{-5} {\mathrm{s}}^{-1}$, which leads to the nondimensional parameter $A_{\kappa }^2=2$. Other parameters used are set to $g=9.8 {\mathrm{m} \mathrm{s}}^{-1}$, $\overline{T}=283 \mathrm{K}$, $T_0=10 °\mathrm{C}$, and $Q_0=\frac{\kappa T_0}{z_0}=0.02 °\mathrm{C}{ \mathrm{m} \mathrm{s}}^{-1}$. Considering the fact that $A_E=0.2236$ and $A_E^4=2.5\times {10}^{-3}\ll 1$, Equation (14) can be further simplified as discussed below.

3.1. Shallow PBL Case

Since $A_E^4\ll 1$, the first two terms in Equation (14) can be directly discarded without losing too much accuracy. Now Equation (14) becomes

$$\frac{{\partial }^{2}T}{\partial z^2}+{\mathrm{Bu}}^2\frac{{\partial }^{2}T}{\partial x^2}-A_{\kappa }^{2}T=0$$

(29)

Note that this is derived from the case in which the Ekman elevation is smaller than the vertical scale of motion, or the shallow PBL case. The higher-order partial differential terms in Equation (14) are ignored in the shallow PBL case so that it can also be called the low-order approximation of Equation (14). It is interesting to note that Equation (29) means that the fourth-order partial differential term in Equation (12) is ignored so that it becomes

$$w=-\frac{gK}{f^2{T}_0}\frac{{\partial }^{2}T}{\partial x^2}$$

(30)

Equation (30) is the thermal wind relationship and demonstrates that the motion Equation (6) in the x-direction takes the form of the geostrophic wind balance (ignoring the eddy viscosity), namely,

$$fv=\frac{1}{{\rho }_0}\frac{\partial p}{\partial x}$$

(31)

which implies that v-wind is balanced by the pressure gradient in the x-direction. The geostrophic wind balance is commonly applied for large-scale motion. Here, it is a natural result for the low-order approximation.

Equation (29) becomes

$$\frac{{\partial }^2{T}_m}{\partial z^2}-\left({\mathrm{Bu}}^2{m}^2+A_{\kappa }^2\right)T_m=0$$

(32)

by applying the Fourier transform, and the solution is

$$T_m=C_1{e}^{-\sqrt{{\mathrm{Bu}}^2{m}^2+A_{\kappa }^2}z}+C_2{e}^{\sqrt{{\mathrm{Bu}}^2{m}^2+A_{\kappa }^2}z}$$

(33)

where $C_1$ and $C_2$ are two integral constants to be determined by the boundary condition. According to the upper boundary condition Equation (3), the temperature is a limited value when $z$ tends to infinity. Therefore, $C_2=0$. According to the lower boundary condition Equation (4), it is easy to derive that $C_1=\frac{q_m}{\sqrt{{\mathrm{Bu}}^2{m}^2+A_{\kappa }^2}}$. Then, Equation (33) becomes

$$T_m=\frac{q_m}{\sqrt{{\mathrm{Bu}}^2{m}^2+A_{\kappa }^2}}{e}^{-\sqrt{{\mathrm{Bu}}^2{m}^2+A_{\kappa }^2}z}$$

(34)

Since $F\left[\frac{e^{-b\sqrt{x^2+a^2}}}{\sqrt{x^2+a^2}}\right]=2K_0\left(a\sqrt{m^2+b^2}\right)$, where $K_0$ is the zeroth-order modified Bessel function of the second kind, the solution of Equation (29) is eventually written as

$$T\left(x,z\right)=\frac{1}{4\pi }\frac{1}{\mathrm{Bu}}{\displaystyle {\int }_{-\infty }^{+\infty }q\left(\xi \right)K_0\left(A_{\kappa }\sqrt{{\left(x-\xi \right)}^2/{\mathrm{Bu}}^2+z^2}\right)d\xi }$$

(35)

The calculation results suggest that the temperature perturbation (Figure 1a) is a typical cooling and heating structure and is antisymmetric at approximately x = 0, analogous to the surface heating function $q\left(x\right)$. The forced u-wind (Figure 1b) is westerly, while the forced v-wind is southerly (Figure 1c). The upwelling (downdraft) is associated with the warm (cold) temperature perturbation (Figure 1d). The forced circulation is strongest at the underlying surface and declines with the vertical height but with no changes in directions. This means that the sinking air in the cooling region blows toward the warming region in the whole layer so that there is no complete circulation cell.

It is obvious that the forced atmospheric circulation is unreasonable compared with the numerical results (Figure 2). Although the temperature perturbation (Figure 2a) is quite similar to that in the previous low-order approximation (Figure 1a), the atmospheric circulation has a large discrepancy. The u-wind (Figure 2b) blows from the cooling region to the warming region in the lower layer (up to approximately 400 m height) and blows inversely in the upper layer. The vertical velocity (Figure 2d) is positive in the warming region and negative in the cooling region, and the strongest upwelling and downdraft are located near 300 m height, rather than the underlying surface, as the low-order approximation suggested in Figure 1d. Now, the sinking air blows toward the warming region in the lower layer, and after rising to a higher level in the warming region, it blows again to the cooling region to form a complete thermal circulation cell.

Now, let us analyze why forced atmospheric motion has a large distortion in the low-order approximation. The vertical velocity and v-wind are calculated according to Equations (30) and(31), the corresponding geostrophic balance and thermal wind relation. They are not precise descriptions for LTCs, which mainly prevail in PBL, where friction is essential and must be considered in any analysis. With this in mind, motion equations should not be further simplified. This means that the viscosity term in Equation (6) should be retained. However, if we take the viscosity term into consideration, we obtain the sixth-order Equation (14). Is there a method that can retain the viscosity in the motion equation and can simplify Equation (14) at the same time? The answer is yes. As long as we replace all the eddy forms of viscosity or damping with a Rayleigh form ($\frac{{\partial }^2}{\partial z^2}\to -\alpha$), we can immediately derive the second-order temperature in Equation (29). In addition, we also inappropriately avoid applying the geostrophic balance and thermal wind relation. In actuality, this is a very common practice in dealing with LTCs (e.g., [1,11,27]). This can also be called the low-order approximation for simplicity. It should be emphasized again that $A_E^4\ll 1$ is essential for the low-order approximation. It is true since the Ekman elevation is smaller than the vertical scale to maintain $A_E=\frac{h_E}{z_0}\sim {10}^{-1}$ in most cases.

3.2. Deep PBL Case

Following the previous discussion, the PBL may be deep compared to the Ekman elevation if $A_E\sim {10}^0$, which means that the Ekman elevation is close to the vertical scale of motion, or the PBL is deep compared to the vertical scale of motion, namely, a deep PBL case. The high-order terms cannot be ignored, and the low-order approximation is invalid. This may be true in some cases. For example, a strong eddy viscosity coefficient (e.g., $\kappa ={100 \mathrm{m}}^2{ \mathrm{s}}^{-1}$) can derive a large Ekman elevation ($h_E=1000 \mathrm{m}$) to make $A_E\sim {10}^0$, and a small vertical scale ($z_0=h_E=223.6 \mathrm{m}$) can also achieve the same thing. To analyze the deep PBL case, $A_E$ is set to unity for simplicity. Since ${\mathrm{Bu}}^2$ also contains $z_0$, it is necessary to shorten the characteristic horizontal scale $L={10}^2{h}_E\sim {10}^4 \mathrm{m}$ to ensure that ${\mathrm{Bu}}^2=1$. Otherwise, ${\mathrm{Bu}}^2\sim {10}^{-2}\ll 1$ means that Equation (14) becomes a sixth-order equation of a single explicit variable $z$.

$$\frac{{\partial }^{6}T}{\partial z^6}-A_{\kappa }^2\frac{{\partial }^{4}T}{\partial z^4}+\frac{{\partial }^{2}T}{\partial z^2}-A_{\kappa }^{2}T=0$$

(36)

Equation (36) only determines the vertical profile of the temperature perturbation. In addition, a stronger Newtonian cooling coefficient ($\tau ={10}^{-3} {\mathrm{s}}^{-1}$) will amplify $A_{\kappa }^2=10$ so that the terms that are not associated with $A_{\kappa }^2$ in Equation (14) can be ignored

$$\frac{{\partial }^{4}T}{\partial z^4}+T=0$$

(37)

Equation (37) has one single variable $z$ and only determines the vertical distribution. The horizontal structure is only determined by the surface heating. On the other hand, a weaker Newtonian cooling coefficient ($\tau ={10}^{-5} {\mathrm{s}}^{-1}$) will make $A_{\kappa }^2=0.1$ so that the terms associated with $A_{\kappa }^2$ in Equation (14) can be ignored:

$$\frac{{\partial }^{6}T}{\partial z^6}+\frac{{\partial }^{2}T}{\partial z^2}+{\mathrm{Bu}}^2\frac{{\partial }^{2}T}{\partial x^2}=0$$

(38)

Although Equation (38) looks simple, it does not take the influence of Newtonian cooling into consideration. To maintain its influence, a moderate Newtonian cooling coefficient ($\tau ={10}^{-4} {\mathrm{s}}^{-1}$) is needed to ensure that $A_{\kappa }^2\sim 1$ so that Equation (14) becomes

$$\frac{{\partial }^{6}T}{\partial z^6}-\frac{{\partial }^{4}T}{\partial z^4}+\frac{{\partial }^{2}T}{\partial z^2}+{\mathrm{Bu}}^2\frac{{\partial }^{2}T}{\partial x^2}-T=0$$

(39)

Equation (39) is equivalent to setting all terms with the same coefficients in Equation (14). Its solution can be obtained by numerically solving the inverse Fourier transform Equation (28). As portrayed in Figure 3a, the vertical temperature profile presents an obvious spiral structure that is caused by the higher-order derivative terms. The negative (positive) temperature anomaly associated with the cooling (warming) surface is limited below approximately 250 m height, turns to positive (negative) anomalies from approximately 250 m to 1300 m height, and turns to negative (positive) anomalies again above the 1300 m height, although the temperature anomaly becomes weaker and tends to be zero. It is interesting to note that the cold (warm) underlying surface can force not only the negative (positive) temperature anomaly but also the positive (negative) anomaly above it. Near the boundary between surface cooling and heating ($\left|x\right|\le 25 \mathrm{km}$), the u-wind (Figure 3b) blows toward the warm region at lower levels and turns its direction to the cold region at middle levels and toward the warm region again at higher levels. Correspondingly, near the boundary region, the air sinks (rises) in the lower cooling (heating) region but rises (sinks) in the upper cooling (heating) region (Figure 3d). This means that the forced atmospheric circulation forms two cells (Figure 4). The lower one is associated with the updrafts in the heating region and the downdrafts in the cooling region. It is a direct thermal circulation. However, the upper one, which shares the same u-wind with the lower one in middle levels, is associated with updrafts in the cooling region and downdrafts in the heating region. It is not a direct thermal cell but a compensatory cell. It is also interesting to note that the sinking air in the cooling region blows not only toward the heating region with a stronger speed to form direct thermal circulation but also toward the cold region ($x<-25 \mathrm{km}$) with a much weaker speed (Figure 3b). This branch of air eventually rises in the cold region (Figure 3c), blows toward the boundary region (Figure 3b) and sinks near to the ground to form a compensatory cell (Figure 4). Similarly, there is also a compensatory cell right of the direct thermal cell. In summary, the direct thermal circulation is bounded in the boundary region and has a smaller vertical height. Three compensatory cells appear to surround the direct cell. This is the most significant discrepancy with the shallow PBL case in the previous subsection. This can be explained by Equation (23), which is the sum of three vertical variation terms. The first term is an exponential decline solution. The latter two terms, however, denote the spiral structure since ${\lambda }_2$ and ${\lambda }_3$ are complex numbers. Therefore, the final solution Equation (23) will have a spiral structure.

Now, let us further discuss the influence of Newtonian cooling. Except for its influence on the magnitude of the temperature perturbation, it can also modulate the temperature distribution. Stronger Newtonian cooling can offset the influence of other terms so that the thermodynamics can determine a spiral vertical distribution of the temperature perturbation (as Equation (37) denoted). Meanwhile, the underlying surface heating $q\left(x\right)$ determines the same horizontal distribution of the temperature perturbation. This means that only the compensatory cell in the vertical direction is retained, but the compensatory cells in the horizontal direction disappear (Figure 5a). On the other hand, weaker Newtonian cooling loses its importance in determining the temperature distribution, as shown in Equation (38) (equivalent to specifying $A_{\kappa }^2=0$ in Equation (14)). Then, the discriminant Equation (21) can be approximately written as

$$\mathsf{\Delta }\approx {\left(\frac{q}{2}\right)}^2$$

(40)

This approximate relation works in most cases, especially when $m^2$ is large. With this approximation, the three roots in Equation (22) reduces to two real roots (the conjugate roots ${\omega }_2$ and ${\omega }_3$ become equal real numbers due to their tiny imagery part). This means that Equation (38) can be reduced to a second order differential equation by ignoring the sixth-order term, namely,

$$\frac{{\partial }^{2}T}{\partial z^2}+{\mathrm{Bu}}^2\frac{{\partial }^{2}T}{\partial x^2}=0$$

(41)

This is the same as the low-order approximation Equation (29) in the shallow PBL case. It is obvious that Equation (41) determines a vertical declining temperature profile with no spiral structure. Therefore, the forced atmospheric circulation is also analogous to that in the shallow PBL case. The compensating circulation cells that surround the direct thermal circulation cell all disappear (Figure 5b). To summarize, a large Newtonian cooling can cause the spiral structure in the vertical direction but lose its influence on the horizontal temperature distribution. Meanwhile, a small Newtonian cooling implies that the results are similar to those in the shallow PBL case. Therefore, moderate Newtonian cooling is critical for compensating circulation cells in the deep PBL case.

4. Summary

This paper investigates the dynamics of LTCs by applying steady, linear, hydrostatic, and incompressible Boussinesq equations. A sixth-order partial differential equation for temperature perturbation can be derived. The equation can be analytically solved by applying the Fourier transform and inverse Fourier transform. It contains three important nondimensional parameters: $A_E$, the ratio between the Ekman elevation and vertical scale of motion; $A_{\kappa }^2$, the ratio between Newtonian cooling and thermal conduction; and $\mathrm{Bu}$, the ratio between the Rossby deformation radius and horizontal scale. The dynamics of LTCs are assessed according to the relative importance of the three parameters.

If the Ekman elevation is smaller than the vertical scale of motion, the PBL depth is relatively shallow compared with the vertical scale of motion. It is defined as the shallow PBL case in which the parameters $A_E<1$ and $A_E^4\ll 1$. Therefore, the higher-order differential terms that contain $A_E^4$ can be directly ignored without losing too much accuracy. The resulting low-order approximation reduces the temperature equation to a second-order one that is easy to solve. Although the viscosity in the motion equation can be ignored in the low-order approximation, directly ignoring viscosity in the motion equation leads to a geostrophic balance that is invalid in the PBL. Therefore, the viscosity should be replaced by a Rayleigh coefficient rather than directly ignored in the low-order approximation. The parameter $\mathrm{Bu}$ controls the horizontal temperature structure. If ${\mathrm{Bu}}^2\ll 1$, the horizontal temperature distribution will only depend on the underlying surface heating. The parameter $A_{\kappa }$ modulates the vertical temperature distribution. The forced atmospheric circulation is a direct thermal circulation cell. The air sinks in the cooling region and blows toward the heating region in the lower level. Then, the air rises in the heating region and blows black to the cooling region at a higher level. Generally, the shallow PBL case is suitable for most LTCs.

If the Ekman elevation is close to or larger than the vertical scale of motion, the PBL depth is relatively deep compared with the vertical scale of motion. It is defined as the deep PBL case, in which $A_E^4\ge 1$, so that the higher-order terms in the temperature equation cannot be ignored. Then, the temperature equation has three characteristic roots, two of which are complex and cause the spiral structure. The corresponding forced atmospheric circulation shows interesting features. In addition to the direct thermal circulation cell that is limited near the boundary region between heating and cooling, there also exist three compensating circulation cells that surround the direct thermal cell. This solution is quite new and has rarely been mentioned in the literature. If the parameter $A_{\kappa }$ is very small, the influence of Newtonian cooling can be ignored. Although this cannot formally simplify the equation, it actually causes the highest-order term to lose its importance so that the temperature equation can also be simplified to a second-order equation, just as the low-order approximation does in the shallow PBL case. Therefore, the spiral structure disappears, and only the direct thermal circulation cell survives. If the Burger number ${\mathrm{Bu}}^2\ll 1$, the horizontal temperature is just a response to the underlying surface heating. This leads to the two compensating circulation cells in the horizontal direction disappearing. However, the compensating cells in the vertical direction can survive due to the obvious vertical spiral structure in the temperature equation. Therefore, a moderate Newtonian cooling coefficient is also critical to reproduce the compensating circulation cells. The deep PBL case requires smaller horizontal and vertical scales of motion than the shallow PBL case.

This paper divides the dynamics of LTCs into two cases. LTCs in the shallow PBL case are more common and have been investigated extensively. However, LTCs in the deep PBL case are rarely discussed. The existence of compensative circulation cells that surround the direct circulation cell is the most important feature. The present study only focuses on the steady state in stationary background wind. In addition, the localizing heating does not appear in the thermodynamic equation but in the lower boundary condition, which may not exhibit its influence directly. These are left to future work.