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  • View in gallery

    An XZ two-dimensional example of the configuration of the IBM gridpoint system. The thick dashed line is the surface of the terrain. The ghost cells are labeled by a letter “G” in red. The grid points in the flow region are displayed by black and blue triangles, with the latter representing these points closest to the terrain surface (also marked by “T”). The circles stand for grid points deep under the terrain, and are not involved in any computation.

  • View in gallery

    The vertical profile of the u component of wind (m s−1) from an idealized experiment. The black line is the model-produced “true” solution, and the red and green lines are the results from WISSDOM synthesis with and without assimilating surface station observations, respectively.

  • View in gallery

    An XZ two-dimensional example of the grid points configuration. The empty circles, crosses, and triangles represent the starting point, middle point, and ending point of a segment in the flow region, respectively, while the filled circles stand for grid points inside the mountain.

  • View in gallery

    At Z = 1.0 km, the model produced (a) horizontal wind (vector) and vertical velocity (color shading in m s−1); (b) pressure perturbation p′ (contour in hPa) and virtual cloud temperature perturbation θc (color shading in K); (c) TPTRS-derived p′ and θc using wind fields. Mountains are represented by black shading.

  • View in gallery

    As in Fig. 4, but for a vertical cross section at Y = 0.0 km. In (a) the arrow denotes the uw winds.

  • View in gallery

    Field of parameter “S” over a vertical cross section at Y = 0.0 km from (a) numerical model and (b) retrieval by TPTRS. The variable has been amplified by 102. See the text for the definition of S.

  • View in gallery

    Surface weather chart at 1200 UTC 14 Jun 2008.

  • View in gallery

    Composite maximum radar reflectivity (color shading in dBZ) at 1500 UTC (2300 LST) 14 Jun 2008.

  • View in gallery

    Accumulated rainfall (color shading in mm) from 1200 to 1500 UTC 14 Jun 2008.

  • View in gallery

    Analysis domain and locations of two CWB S-band Doppler radars (brown triangle), NCAR S-POL (brown diamond), profiler (red square), PingTung radiosonde station (blue circle), LiouGuei radiosonde station (green circle), and surface stations (purple cross). The gray shading denotes the terrain heights. The thick solid line is the coastline.

  • View in gallery

    Horizontal wind at 1500 UTC from (a) retrieval by WISSDOM and (b) profiler observations. The color shading represents the wind speed in m s−1.

  • View in gallery

    Observed (solid lines) and TPTRS-retrieved (dashed lines) pressure perturbation (p′ in hPa) and virtual cloud temperature perturbation (θc in K). Black and red lines indicate the pressure and temperature, respectively. The comparisons are conducted using data from (a) PingTung and (b) LiouGuei radiosondes, respectively.

  • View in gallery

    Composite maximum radar reflectivity (color shading in dBZ), horizontal winds (arrow), and horizontal divergence (contour in s−1) at the height 0.5 km. Positive and negative divergence are depicted by the solid and dashed lines, respectively. The gray shading denotes mountainous areas with altitudes greater than 0.5 km. The thick solid line is the coastline.

  • View in gallery

    As in Fig. 13, but for retrieved pressure perturbation (p′, contour in hPa) and virtual cloud temperature perturbation (θc, color shading in K). The rectangle enclosed by the green line denotes the area in which the vertical structure of the retrieved thermodynamic fields will be discussed in Fig. 15.

  • View in gallery

    The storm-relative uw wind field (arrow), radar reflectivity (color shading in dBZ), and vertical velocity (contour in m s−1) over a vertical cross section for the area outlined in Fig. 14. The parameters displayed in this figure are averaged over a horizontal distance of 20 km. The mountain is denoted by the filled gray shading.

  • View in gallery

    As in Fig. 15, but for the retrieved pressure perturbation (p′, contour in hPa) and virtual cloud temperature perturbation (θc, color shading in K).

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Thermodynamic Recovery of the Pressure and Temperature Fields over Complex Terrain Using Wind Fields Derived by Multiple-Doppler Radar Synthesis

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  • 1 Department of Atmospheric Sciences, National Central University, Zhongli, Taoyuan City, Taiwan
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Abstract

A new thermodynamic retrieval scheme is developed by which one can use the wind fields synthesized from multiple-Doppler radars to derive the three-dimensional thermodynamic fields over complex terrain. A cost function consisting of momentum equations and a simplified thermodynamic equation is formulated. By categorizing the analysis domain into flow and terrain regions, the variational technique is applied to minimize this cost function only within the flow region, leading to the solutions for the three-dimensional pressure and temperature perturbations immediately over terrain. Using idealized datasets generated by a numerical model, an experiment is first conducted to assess the accuracy of the proposed algorithm. The retrieval scheme is then applied to a real case that occurred during the 2008 Southwestern Monsoon Experiment (SoWMEX) conducted in Taiwan. The retrieved thermodynamic fields, verified by radiosonde data, reveal the structure of a prefrontal squall line as it approaches a mountain. The retrieved three-dimensional high-resolution pressure and temperature along with the wind fields not only allow us to better understand the thermodynamic and kinematic structure of a heavy rainfall system, but can also be assimilated into a numerical model to improve the forecast.

© 2019 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Yu-Chieng Liou, tyliou@atm.ncu.edu.tw

Abstract

A new thermodynamic retrieval scheme is developed by which one can use the wind fields synthesized from multiple-Doppler radars to derive the three-dimensional thermodynamic fields over complex terrain. A cost function consisting of momentum equations and a simplified thermodynamic equation is formulated. By categorizing the analysis domain into flow and terrain regions, the variational technique is applied to minimize this cost function only within the flow region, leading to the solutions for the three-dimensional pressure and temperature perturbations immediately over terrain. Using idealized datasets generated by a numerical model, an experiment is first conducted to assess the accuracy of the proposed algorithm. The retrieval scheme is then applied to a real case that occurred during the 2008 Southwestern Monsoon Experiment (SoWMEX) conducted in Taiwan. The retrieved thermodynamic fields, verified by radiosonde data, reveal the structure of a prefrontal squall line as it approaches a mountain. The retrieved three-dimensional high-resolution pressure and temperature along with the wind fields not only allow us to better understand the thermodynamic and kinematic structure of a heavy rainfall system, but can also be assimilated into a numerical model to improve the forecast.

© 2019 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Yu-Chieng Liou, tyliou@atm.ncu.edu.tw

1. Introduction

Doppler radar can detect the wind field with high spatiotemporal resolution, but only along the radial direction. Armijo (1969) first demonstrated that it is mathematically feasible to use data collected by two or three Doppler radars to uniquely determine the wind velocity and terminal fall speed of the raindrops. Ray et al. (1975) provided the first three-dimensional wind fields in a tornadic storm using observations from two Doppler radars. Over the past few decades, numerous studies about dual-Doppler or multiple-Doppler wind synthesis techniques have been conducted, allowing the estimation of the error distribution of the derived wind field (Doviak et al. 1976), improved treatment of the top or bottom boundary conditions for the vertical velocity (O’Brien 1970), development of an iterative approach to find the solution for the wind field (Brandes 1977), implementation of variational adjustment (Ray et al. 1980; Ziegler et al. 1983), and so on. New analytical schemes using multiple-Doppler radar measurements have also been proposed by Scialom and Lemaitre (1990), Protat and Zawadzki (1999), and Shapiro and Mewes (1999), among others.

However, a Doppler weather radar is not able to directly observe the thermodynamic parameters such as temperature and pressure. These variables, especially their three-dimensional distribution, are useful for various applications, such as studies of the evolution and triggering mechanisms of mesoscale or storm-scale weather systems, and improvement of numerical model forecast skill through data assimilation. This has made the so-called thermodynamic retrieval technique, a procedure by which one can derive three-dimensional pressure and temperature fields from multiple-Doppler radars synthesized winds, an important research topic in the radar meteorology community. The method developed first by Gal-Chen (1978) was particularly applicable, because the boundary condition for solving a Poisson equation to obtain the pressure perturbation could naturally be determined by the wind fields synthesized by Doppler radars. This method was later widely adopted to study the structure of various weather systems such as a deep moisture convection (Hane et al. 1981), a dry boundary layer (Gal-Chen and Kropfli 1984), a tornadic thunderstorm (Hane and Ray 1985), convection embedded in a squall line (Lin et al. 1986), and a frontal rainband (Parsons et al. 1987). To investigate the scale interactions, Protat et al. (1998) also introduced a new procedure for recovering the thermodynamic fields from Doppler radar data.

However, it should be pointed out that the products retrieved using the aforementioned algorithms only represent the deviation of the pressure and temperature perturbations (defined as the difference with respect to a base state) from their horizontal average at each layer. Fortunately, this limitation does not affect the structure of the thermodynamic fields over a horizontal plane. On the other hand, a correct interpretation of the vertical distribution of the pressure and temperature perturbations is still possible if the weather system is well restricted inside the computational domain and the forcing is sufficiently insignificant along the boundaries (e.g., Brandes 1984). However, when this condition is not satisfied and the vertical structure of the thermodynamic field is needed, as pointed out by Gal-Chen (1978), it is necessary to have at least one independent field measurement of the temperature and pressure for each altitude, which may not always be available.

Roux (1985, 1988) proposed a new approach in which a unique solution of the pressure and temperature perturbations could be achieved up to a volume-wide constant. This constant can be deduced from one independent pressure and temperature measurement at a single point in the domain. Sun and Roux (1988) applied the results obtained by Roux (1985, 1988) to investigate the trailing anvil clouds of squall lines over a vertical cross section. Roux and Sun (1990) and Roux et al. (1993) made further improvements of the Roux (1988) scheme by including a thermodynamic equation along with the equations of motion in their retrieval scheme, so that the temperature gradient could be provided in any direction. Sun and Crook (1996) compared the 4D-Var adjoint technique and the traditional Gal-Chen (1978) scheme. The advantage of the 4D-Var formulation was that one could deduce the three-dimensional thermodynamic fields without extra and independent measurements.

However, all of the retrieval schemes mentioned above are only applicable to recovering the thermodynamic fields over a flat surface. It is known that complex terrain will significantly complicate the computation. For example, in Gal-Chen (1978), a Poisson equation is solved for each horizontal domain to obtain the pressure perturbation. If complex topography exists, the horizontal domain that intersects the mountains will contain hollows with irregular boundaries. Solving the Poission equation over such a domain becomes very complicated (Maury 2001). In recent years, the first author of this manuscript developed a new multiple-Doppler radar wind synthesis algorithm, named the Wind Synthesis System using Doppler Measurements (WISSDOM; see Liou et al. 2012 and Liou et al. 2014). By employing the so-called Immersed Boundary Method to compute the forcing exerted by the terrain on the fluid, WISSDOM was able to synthesize the three-dimensional wind fields over complex terrain using multiple-Doppler radar observations. To fully take advantage of WISSDOM, in this current work a new method, named Terrain-Permitting Thermodynamic Retrieval Scheme (TPTRS), is developed whereby one can use the wind fields synthesized by WISSDOM to derive the three-dimensional thermodynamic fields immediately over complex terrain.

The rest of the manuscript is organized as follows. Section 2 gives an introduction to WISSDOM and TPTRS. The performance of TPTRS is tested in section 3. An example of the application of this scheme to recover the thermodynamic fields for a real case is presented in section 4, followed by some conclusions in section 5.

2. Methodology

a. WISSDOM: A multiple-Doppler radar three-dimensional wind synthesis method over complex terrain

In this study, an advanced algorithm (named WISSDOM) is employed to conduct wind field analysis. As introduced in Liou et al. (2012), the wind fields are solved by variationally minimizing a cost function, which includes a set of weak constraints representing the multiple-radar Doppler radial wind observations, anelastic continuity equation, vertical vorticity equation, background flow field, and spatial smoothness terms. The background wind field is utilized to fill in the data void region, and can be provided by the outputs from a mesoscale numerical model (Liou et al. 2014). Compared with other commonly used traditional approaches, this method has several advantages. For example, it is able to recover the wind field along the radar baseline (Liou and Chang 2009) and immediately above complex terrain (Liou et al. 2012). The latter is accomplished by employing the immersed boundary method (IBM, to be explained below) to handle the bottom boundary conditions, allowing topographic forcing on the fluid to be considered during the wind synthesis. Data from any number of radars can be easily merged (Liou et al. 2016). In addition, the vertical vorticity equation is implemented to constrain the retrieved three-dimensional winds, thus the production of a residual term when performing the vorticity budget analysis can be avoided (Liou et al. 2012). Protat and Zawadzki (2000) also pointed out that a wind field that satisfies the vertical vorticity equation, if applied to thermodynamic retrieval, could yield results with a higher accuracy. Applications of WISSDOM for studying the convection structure or orographic effects can be found in Liou et al. (2013), Lee et al. (2014), Chang et al. (2015), Liou et al. (2016), Lee et al. (2018), and Tsai et al. (2018).

Figure 1 illustrates the configuration of the grid system in IBM. The grid points below the terrain but nearest to the terrain surface are defined as the ghost cells (Tseng and Ferziger 2003). The winds at these so-called ghost cell points are obtained through an interpolation using data at neighboring grid points located in the flow region. The interpolation is designed so that certain boundary conditions for the flow fields can be satisfied along the terrain surface (Liou et al. 2012). As a result, when computing the gradients of the flow fields at these grid points located in the flow region and closest to the terrain surface (marked by a letter T and displayed by blue triangles in Fig. 1), the wind information needed at nearby ghost cell points would be available.

Fig. 1.
Fig. 1.

An XZ two-dimensional example of the configuration of the IBM gridpoint system. The thick dashed line is the surface of the terrain. The ghost cells are labeled by a letter “G” in red. The grid points in the flow region are displayed by black and blue triangles, with the latter representing these points closest to the terrain surface (also marked by “T”). The circles stand for grid points deep under the terrain, and are not involved in any computation.

Citation: Monthly Weather Review 147, 10; 10.1175/MWR-D-19-0059.1

Given that most radar scans are made from a positive elevation angle and Earth’s surface is curved, the atmosphere at lower levels is usually undetectable by a radar. In this study WISSDOM is further improved by merging the wind fields observed by surface stations in order to obtain information about the flow fields near the ground, where the radar data are usually not available.

Thus, an additional constraint is added to the cost function of WISSDOM with the following form:
Jsfc=αsfc(AA¯sfc)2,
where A can be horizontal wind components u or υ at a grid point, A¯sfc is the targeted value (explained below) toward which parameter A is adjusted, and the weighting coefficient αsfc is given a value of 500. Note that Jsfc is applied only to the analysis level with available information provided by surface stations. This is accomplished by moving the surface station data vertically to the nearest analysis level using a method described later in this section.
A¯sfc is a weighted average expressed by
A¯sfc=i=1Nwti,new×Astn,i,
where Astn denotes the value of A from surface station observation, N is the total number of applicable surface stations, i is the index of the station, and wti,new is a normalized weighting coefficient explained in the following.
Each surface station is given a two-dimensional Gaussian-type weighting function (wt) defined by
wt=exp{12σ2[(xxs)2+(yys)2]},
in which (x, y) and (xs, ys) are the coordinates of a grid point and a surface station, respectively, and σ is specified to be 5.0 km. A cutoff value of 0.5 is specified, meaning the parameter at a given grid point is adjusted by a surrounding surface station only when the weighting wt from this particular station is greater than 0.5. If a grid point is influenced by more than one nearby surface station, the weighting coefficients from each station need to be normalized by
wti,new=wtij=1Nwtj,
where i is the index of the station, and wti (or wtj) and wti,new are the weighting coefficients before and after the normalization, respectively.
However, if a grid point is within the radius of influence of only one nearby surface station, the A¯sfc at this point will be
A¯sfc=wt×Astn+(1wt)Abgrd,
where Abgrd stands for the value of A from the background field, which can be obtained from reanalysis data or the output of a mesoscale model.
Usually the altitudes of the stations do not match the WISSDOM analysis levels exactly, especially for those stations deployed in mountainous areas. Under this condition, the observed u and υ at the surface stations are shifted vertically to the nearest analysis level through an empirical power law as suggested by Peterson and Hennessey (1978):
windstnwindWISSDOM=(HstnHWISSDOM)P,
where windstn and windWISSDOM represent the u or υ wind component observed by the station at the altitude Hstn, and the extrapolated results located at the WISSDOM analysis levels HWISSDOM, respectively. According to Hsu et al. (1994) and Chen et al. (2016), P is specified to be 0.143. Finally, for those grid points located between the lowest radar observational level and the surface, a penalty term, shown in (7), is implemented so as to adjust the wind fields toward the background values that are obtained through a linear interpolation using the wind information below and aloft:
Jimiddle=αmiddle{Ai[(ZiZsZrZs)Ar+(ZrZiZrZs)As]}2,
where Ar and As are the u or υ winds at the grid points located on the lowest analysis level (Z = Zr) with available radar data and the surface level (Z = Zs), respectively, and Ai is the wind field located at these grid points in between (Z = Zi). The radar scans in Taiwan are often blocked by the Central Mountain Range (CMR), whose height is roughly 3.0 km. In addition, the applicable range of the linear interpolation should not be too long. Thus, (7) is utilized to fill in the low-level data gap only when the height of the available radar data Zr is less than 3.0 km. The coefficient αmiddle is given a value of 100.

Figure 2 shows the result from an idealized experiment in which the “true” wind field is from a numerical model simulation. It can be seen that by using the procedure introduced in this section, WISSDOM is able to use the surface station observations to provide a better description of the wind field in low levels where the radar observations are usually unavailable.

Fig. 2.
Fig. 2.

The vertical profile of the u component of wind (m s−1) from an idealized experiment. The black line is the model-produced “true” solution, and the red and green lines are the results from WISSDOM synthesis with and without assimilating surface station observations, respectively.

Citation: Monthly Weather Review 147, 10; 10.1175/MWR-D-19-0059.1

Since WISSDOM is able to provide three-dimensional wind fields over complex terrain, in the following section TPTRS is introduced so that one can directly use the WISSDOM-produced wind fields to recover the thermodynamic parameters over complex terrain.

b. A Terrain-Permitting Thermodynamic Retrieval Scheme (TPTRS)

Our newly designed thermodynamic retrieval scheme utilizes the same set of basic equations of motion as in Liou et al. (2003), and includes the contributions from moisture. They are expressed as follows:
1θυ0[ut+Vufυ+turb(u)]=πxF;
1θυ0[υt+Vυ+fu+turb(υ)]=πyG;
1θυ0[wt+Vw+turb(w)+g(qr+qs)]=πz+gθcθυ0θ0H,
where the subscript ‘‘0’’ represents a horizontally homogeneous basic state, which can be determined by an environmental sounding. The nonhydrostatic perturbations from the basic state are expressed by variables with a prime. In (8)(10), the (u, υ, w) denote the Cartesian wind components, f is the Coriolis parameter, g stands for the gravity, and turb( ) represents a subgrid-scale turbulence parameterization operator, which are parameterized using a simple first-order closure scheme in this study. The mixing ratio of rainwater (qr) and snow (qs) can be estimated using the radar reflectivity data through empirical equations (e.g., Tong and Xue 2005). The variable π stands for a normalized pressure called the Exner function, and is defined as follows:
π=Cp(PP0)R/Cp,
where P is the pressure, P0 equals 100 kPa, R is the gas constant, and Cp refers to the specific heat capacity at a constant pressure. The virtual potential temperature (θυ) and virtual cloud potential temperature perturbation (θc, see Roux 1985) are defined by
θυ=θ(1.0+0.61qυ),
θc=θ+(0.61qυqc)θ0,
where θ is potential temperature, qυ stands for the perturbation of the water vapor mixing ratio, and qc is the cloud water mixing ratio. The virtual cloud potential temperature perturbation (θc) is treated as a retrievable parameter in the proposed formulation.

The values of F, G, and H can be obtained once the three-dimensional air motion is obtained by WISSDOM.

In addition to the momentum equations, a simplified thermodynamic equation is also employed for the virtual cloud temperature perturbation (θc):
uθcx+υθcy+wθcz+wdθ0dz+S=0,
where S stands for the total effect from the temporal variation, diffusion, and the source/sink of θc obtained through microphysical processes. Similar to the experiments detailed in Liou (2001), the S term is treated as a retrievable parameter in this study, and no additional parameterizations are applied.
A cost function is formulated using the equations listed in (8)(10) and (14), as follows:
J=12i=14Ji,
J1=xyzα1[πxF]2=xyzα1(C1)2,
J2=xyzα1[πyG]2=xyzα2(C2)2,
J3=xyzα3[πzgθcθ0θυ0H]2=xyzα3(C3)2,
J4=xyzα4[uθcx+υθcy+wθcz+wdθ0dz+S]2=xyzα4(C4)2.

The weighting coefficients α1α4 are used to balance the contribution from each term. They are determined following the principles discussed in Liou (2001).

As described in Liou (2001), in order to minimize the cost function J expressed in (15a) in a three-dimensional space, the quasi-Newtonian conjugate-gradient algorithm (Liu and Nocedal 1988) is employed. This method requires information about the cost function gradients with respect to the control variables (i.e., π′, θc, S) at each grid point. To perform the minimization immediately above the terrain, Fig. 3 shows that for a given straight line connecting either the lateral or top/bottom boundaries of the analysis domain, it is divided by terrain into segments. The grid points along each segment are categorized into interior, starting, and ending points. A starting (ending) point is the grid point nearest to a mountain of its right (left) in the x and y directions. As for the vertical direction, the starting point is the first grid above the mountain, while the ending point is usually at the top of the analysis domain. The gradients in each category are derived separately.

Fig. 3.
Fig. 3.

An XZ two-dimensional example of the grid points configuration. The empty circles, crosses, and triangles represent the starting point, middle point, and ending point of a segment in the flow region, respectively, while the filled circles stand for grid points inside the mountain.

Citation: Monthly Weather Review 147, 10; 10.1175/MWR-D-19-0059.1

For interior points:
Jπ=[(α1C1)x+(α2C2)y+(α3C3)z]
Jθc=gα3C3θ0θυ0[(α4uC4)x+(α4υC4)y+(α4wC4)z]
JS=α4C4.
For starting/ending points along the x direction:
Jπ=α1C1
Jθc=α4uC4
JS=0,
where the minus (plus) sign indicates the gradients located at the starting (ending) boundary points of this segment along the x direction.
For starting/ending points along the y direction:
Jπ=α2C2
Jθc=α4υC4
JS=0,
where the minus (plus) sign indicates the gradients located at the starting (ending) boundary points of this segment along the y direction.
For starting/ending points along the z direction:
Jπ=α3C3
Jθc=α4wC4
JS=0,
where the minus (plus) sign denotes the gradients located at the starting (ending) boundary points of this segment along the z direction, which are equivalent to the bottom (top) boundary points in the vertical direction.

It should be mentioned that according to the definition described in this section, when conducting thermodynamic retrieval, the grid points marked by “T” in Fig. 1 would be the ending points along the x direction, and starting points along the z direction. The retrieval is performed only at these grid points located in the flow region (marked by black and blue triangles in Fig. 1). The IBM is only utilized in WISSDOM to provide boundary conditions for wind fields. It is not applied in TPTRS.

Starting from an initial guess for the thermodynamic fields, along with the gradients computed using (16)(19), an iterative process is performed to minimize the cost function shown in (15). Since (15) is formulated using data only at these grid points located in the flow region, the solution yielded by TPTRS is the thermodynamic field in a three-dimensional space with its bottom boundary following the surface of the terrain.

3. Thermodynamic retrieval results from OSSE tests

An observing system simulation experiment (OSSE) is first conducted to investigate the performance of this newly developed thermodynamic retrieval system. The “observations” are provided by a simulation conducted using the Weather Research and Forecasting (WRF) Model. The simulated three-dimensional wind components and rainwater mixing ratio are utilized to generate pseudo radar observational datasets.

Idealized bell-shaped mountains expressed by the following formula are used to represent the orography:
H(x,y)=h[(xxca)2+(yycb)2+1]1.5,
where h is the crest height, (xc, yc) stands for the center of the mountain, and a and b are the mountain half-width along the east–west, north–south directions. Two mountains are placed inside the model domain centered at (23, 20 km), (47, 20 km), with the peak height set to be 1.5 and 2.5 km, respectively. The mountain half widths along all directions for both mountains are specified to be 5.0 km. To initiate the simulation, thermal bubbles are placed in the model domain by superimposing the perturbation (θ′) to the potential temperature field using the following formula:
θ=θmaxcos2(π2Rad),
Rad=(xx04.0)2+(yy04.0)2+(zz01.5)2,
where (x0, y0, z0) represent the center of the perturbation. Note that (21) is applied only when Rad ≤ 1. The storm starts from two thermal bubbles centered at (12, 20 km), (38, 20 km), with the maximum magnitudes θmax reaching 0.8 and 1.0 k, respectively. The model results are interpolated to a domain covering an area of 70 km × 40 km × 15 km, with the horizontal and vertical resolutions being 1.0 and 0.5 km, respectively. The model-simulated wind fields (u, υ, w) along with rainwater (qr) are plugged into (8)(10) to generate F, G, and H, which are then used as inputs for (15). Equation (15) is variationally minimized to produce a set of optimally determined pressure and temperature fields (π′, θc). Since the purpose of this idealized case test is to check the accuracy of the algorithm, the data generated by the model are applied without adding any error.

Figure 4 shows the horizontal and vertical wind fields, the “true” and retrieved thermodynamic fields by TPTRS on a horizontal plane at Z = 1.0 km. At this height, as illustrated in Fig. 4a, an updraft and downdraft occur on the windward and leeside of both mountains, which are associated with an accumulation of cold and warm air, respectively, as shown in Fig. 4b. Negative pressure perturbations are found surrounding the mountains. Figure 4c shows that with correct wind information the retrieved thermodynamic fields are in good agreement with the true solutions not only qualitatively, but also quantitatively.

Fig. 4.
Fig. 4.

At Z = 1.0 km, the model produced (a) horizontal wind (vector) and vertical velocity (color shading in m s−1); (b) pressure perturbation p′ (contour in hPa) and virtual cloud temperature perturbation θc (color shading in K); (c) TPTRS-derived p′ and θc using wind fields. Mountains are represented by black shading.

Citation: Monthly Weather Review 147, 10; 10.1175/MWR-D-19-0059.1

Figure 5 illustrates the wind field, model-generated “true” thermodynamic fields over an east–west-oriented vertical cross section penetrating the center of the domain at Y = 0 km, as well as their retrieved counterparts. It can be seen clearly from Fig. 5a, that the airflow is lifted by the mountains. Figure 5b shows that positive temperature perturbations, originating from the thermal bubbles, occupy the windward side of the mountains above the height Z ~1.0 km, and can also be found above the mountain peaks. They are associated with negative pressure perturbations. Along the top periphery of the warm regions, one can identify positive pressure perturbations. It is known that the buoyancy field (B), which can be approximated by the temperature distribution, also contributes to the generation of the pressure perturbations. According to Markowski and Richardson (2011), we have
Pb(Bz),
where Pb represents the buoyancy-induced pressure perturbation. Since the vertical gradient of B is negative along the top boundary of the warm regions, thus one can use (22) to reasonably explain the existence of the positive pressure perturbations.
Fig. 5.
Fig. 5.

As in Fig. 4, but for a vertical cross section at Y = 0.0 km. In (a) the arrow denotes the uw winds.

Citation: Monthly Weather Review 147, 10; 10.1175/MWR-D-19-0059.1

Figure 5c depicts the retrieved temperature and pressure perturbation fields over the mountainous area obtained using the wind information. The results indicate that the aforementioned thermodynamic features adjacent to the mountains are accurately retrieved by the proposed algorithm. Figure 6 displays the true and retrieved residual terms (S), as defined in (14). Similar to the experimental results reported in Liou (2001), the retrieved S term agrees well with its true counterpart. Nevertheless, the successful recovery of the residual term over terrain is particularly encouraging. It is pointed out in Liou (2001) that this variable represents the total effects of the local temporal change, turbulence, and source/sink to the temperature. Therefore, if further parameterization to separate each term’s contribution is performed, the retrieved S can be used as a guide to constrain the total amount.

Fig. 6.
Fig. 6.

Field of parameter “S” over a vertical cross section at Y = 0.0 km from (a) numerical model and (b) retrieval by TPTRS. The variable has been amplified by 102. See the text for the definition of S.

Citation: Monthly Weather Review 147, 10; 10.1175/MWR-D-19-0059.1

A quantitative comparison is obtained by computing the spatial correlation coefficient (SCC) and root-mean-square errors (RMSEs) between the true and retrieved fields over the three-dimensional domain. The SCC values for the pressure and temperature are 0.96 and 0.94, while the RMSEs are 0.03 hPa and 0.06 K, respectively. These statistics indicate a successful recovery of the thermodynamic fields over terrain using wind information.

4. Thermodynamic retrieval results from a real case study

a. 2008 SoWMEX field experiment and verification

The Southwestern Monsoon Experiment (SoWMEX) field experiment was conducted in Taiwan from May to June 2008. Its scientific goal was to explore the mechanisms leading to the heavy rainfall in Taiwan and the vicinity that occurs during the Asian summer monsoon season (Jou et al. 2011). It is expected that the knowledge obtained from this experiment can be applied to improve model forecasts of precipitation.

The case selected for this study took place during intensive observing period (IOP) 8 at 1500 UTC 14 June 2008. Figure 7 shows that on that day, a stationary front was located to the northwest of Taiwan, and the composite radar reflectivity displayed in Fig. 8 suggests a prefrontal squall line system. Figure 9 shows the accumulated rainfall generated by Taiwan Central Weather Bureau (CWB) Quantitative Precipitation Estimation and Segregation Using Multiple Sensor (QPESUMS) system (Zhang et al. 2008), which basically represents the radar-measured precipitation calibrated by rain gauge observations. Figure 9 reveals a northeast–southwest-oriented rainband that extended from the ocean to the land in southern Taiwan. A wide range of heavy precipitation can be found over the land, with the 3-h rainfall accumulation from 1200 to 1500 UTC exceeding 50 mm. In some areas of the southern plain of Taiwan more than 200 mm of rainfall fell within 24 h (not shown).

Fig. 7.
Fig. 7.

Surface weather chart at 1200 UTC 14 Jun 2008.

Citation: Monthly Weather Review 147, 10; 10.1175/MWR-D-19-0059.1

Fig. 8.
Fig. 8.

Composite maximum radar reflectivity (color shading in dBZ) at 1500 UTC (2300 LST) 14 Jun 2008.

Citation: Monthly Weather Review 147, 10; 10.1175/MWR-D-19-0059.1

Fig. 9.
Fig. 9.

Accumulated rainfall (color shading in mm) from 1200 to 1500 UTC 14 Jun 2008.

Citation: Monthly Weather Review 147, 10; 10.1175/MWR-D-19-0059.1

Data observed by three S-band radars are utilized. They are the S-band dual-polarization Doppler radar (S-POL), operated by the National Center for Atmospheric Research (NCAR), and RCCG and RCKT operated by the CWB of Taiwan. The latter two are Doppler radars. Figure 10 depicts the locations of these three radars, surface stations, along with wind profiler, PingTung and LiouGuei radiosonde stations from which the observational data are adopted for verification.

Fig. 10.
Fig. 10.

Analysis domain and locations of two CWB S-band Doppler radars (brown triangle), NCAR S-POL (brown diamond), profiler (red square), PingTung radiosonde station (blue circle), LiouGuei radiosonde station (green circle), and surface stations (purple cross). The gray shading denotes the terrain heights. The thick solid line is the coastline.

Citation: Monthly Weather Review 147, 10; 10.1175/MWR-D-19-0059.1

Figure 11 shows a comparison of the retrieved horizontal wind against the wind profiler measurements from the surface to Z = 8 km. It can be seen from the WISSDOM-retrieved results (Fig. 11a) that the wind direction is southerly at the lower layers, but shifts to southwesterly and westerly at the middle and upper layers, respectively. The wind speed varies within the range of 5.0 and 15.0 m s−1. The vertical distribution of the retrieved wind field at this location is verified satisfactorily by the independent wind profiler observations as illustrated in Fig. 11b.

Fig. 11.
Fig. 11.

Horizontal wind at 1500 UTC from (a) retrieval by WISSDOM and (b) profiler observations. The color shading represents the wind speed in m s−1.

Citation: Monthly Weather Review 147, 10; 10.1175/MWR-D-19-0059.1

Figure 12 displays the profiles of pressure and temperature perturbations up to Z = 5.0 km observed by the PingTung and LiouGuei radiosondes and their retrieved counterparts. The observations indicate that the pressure perturbation is rather weak, while the temperature perturbation is generally negative at lower layers, but becomes positive at higher altitudes, varying from −1.5 to +1.5 K. These features are reasonably recovered by TPTRS. It should be pointed out since the locations of the radiosondes do not match the grid points exactly. Thus, the grid point closest to the radiosonde is selected and the comparison is conducted using the retrieved temperature and pressure right above this grid. The horizontal drift of the radiosonde during its ascending process is not considered. We believe these factors may contribute to the differences between the retrieved and observed profiles. Nevertheless, the retrieved profiles have similar orders of magnitude and trends as the observations.

Fig. 12.
Fig. 12.

Observed (solid lines) and TPTRS-retrieved (dashed lines) pressure perturbation (p′ in hPa) and virtual cloud temperature perturbation (θc in K). Black and red lines indicate the pressure and temperature, respectively. The comparisons are conducted using data from (a) PingTung and (b) LiouGuei radiosondes, respectively.

Citation: Monthly Weather Review 147, 10; 10.1175/MWR-D-19-0059.1

b. Interpretation of the retrieved thermodynamic field

This section discusses the structure of the TPTRS-retrieved thermodynamic fields using WISSDOM-synthesized wind fields. Note that the kinematic and thermodynamic fields are obtained with a horizontal and vertical resolution of 1.0 and 0.25 km, respectively. Figure 13 depicts the horizontal velocity and divergence field at Z = 0.5 km, and the column vector (CV) radar reflectivity (defined as the composite maximum reflectivity of each column). The prevailing wind blows from the ocean to the southwestern plain of the island, but is deflected as it approaches the CMR. A northeast–southwest-oriented band of intensive reflectivity can be clearly identified, within which one finds strong convergence. Based on the assumption of radar reflectivity conservation, Liou and Luo (2001) and Liou (2007) developed a method to objectively determine the moving speed of a weather system. By adopting this method, it is found that the squall line system is moving roughly toward the east, with a direction 255.8° and at a speed of 13.3 m s−1.

Fig. 13.
Fig. 13.

Composite maximum radar reflectivity (color shading in dBZ), horizontal winds (arrow), and horizontal divergence (contour in s−1) at the height 0.5 km. Positive and negative divergence are depicted by the solid and dashed lines, respectively. The gray shading denotes mountainous areas with altitudes greater than 0.5 km. The thick solid line is the coastline.

Citation: Monthly Weather Review 147, 10; 10.1175/MWR-D-19-0059.1

Figure 14 shows the pressure and temperature perturbation fields obtained by TPTRS. A northwest–southeast-oriented band of negative temperature perturbation (cold pool), roughly perpendicular to the moving direction of the squall line, is obtained along the coast. The northern part of the cold pool is produced by the passage of the precipitating convection when moving from the ocean to the land, while the southern part of the cold pool, with a stronger intensity than the northern part, is triggered by the precipitation that has already existed for several hours near the coast (see Fig. 9). The intensity of the cold pool reaches −3.5 K. There is a positive temperature perturbation over the southwest plain area and near the foothills of the CMR. It should also be noticed that over the ocean, the southern section of the rainband (X ~ 70 km, Y = 0 ~ 40 km) is found to be at the same location as the positive temperature perturbation, implying that convection driven by warmer air is still in the early stage of development. Finally, positive (negative) pressure perturbation is generally associated with cold (warm) region.

Fig. 14.
Fig. 14.

As in Fig. 13, but for retrieved pressure perturbation (p′, contour in hPa) and virtual cloud temperature perturbation (θc, color shading in K). The rectangle enclosed by the green line denotes the area in which the vertical structure of the retrieved thermodynamic fields will be discussed in Fig. 15.

Citation: Monthly Weather Review 147, 10; 10.1175/MWR-D-19-0059.1

Figure 15 depicts the storm-relative uw wind field and radar reflectivity over a vertical cross section. The orientation of this vertical cross section is parallel to the motion of the system, and the variables displayed are the mean results from an average over a 20 km horizontal distance (see Fig. 14). The wind field exhibits a typical squall line structure, with evident front-to-rear and rear-to-front flows. The leading edge is located at approximately X = 65 km, where the updraft can reach 2.0 ms−1. On the western side of the CMR (X ~ 110 km), upward motion is triggered by the orographic lifting, while downward motion can be identified on the eastern side of the CMR (X ~ 140 km).

Fig. 15.
Fig. 15.

The storm-relative uw wind field (arrow), radar reflectivity (color shading in dBZ), and vertical velocity (contour in m s−1) over a vertical cross section for the area outlined in Fig. 14. The parameters displayed in this figure are averaged over a horizontal distance of 20 km. The mountain is denoted by the filled gray shading.

Citation: Monthly Weather Review 147, 10; 10.1175/MWR-D-19-0059.1

Figure 16 displays the retrieved thermodynamic fields over the same vertical cross section. A cold pool is identified near the ground extending from X ~ 60 to X ~ 120 km. It is triggered by evaporative cooling of the rainband (see Fig. 9) and is associated with a positive pressure perturbation. An area of positive temperature perturbation, residing from X ~ 0 km to X ~ 40 km, is caused by the transportation of warm air due to the persistent southwesterly flow, leading to a pressure deficit greater than −0.60 hPa. Regions with local minimum pressure can be found at (X ~ 85 km, Z ~ 7 km), while local maximum pressure can be identified at (X ~ 75 km, Z ~ 6 km) and (X ~ 145 km, Z ~ 3 km). They are collocated with the top boundary of regions with temperature minimum and maximum, respectively. These features can be explained by the equation shown in (22).

Fig. 16.
Fig. 16.

As in Fig. 15, but for the retrieved pressure perturbation (p′, contour in hPa) and virtual cloud temperature perturbation (θc, color shading in K).

Citation: Monthly Weather Review 147, 10; 10.1175/MWR-D-19-0059.1

5. Summary

A new Terrain-Permitting Thermodynamic Retrieval Scheme (TPTRS) is developed, which can be used to retrieve the three-dimensional pressure and temperature fields immediately over complex terrain using wind information synthesized from multiple Doppler radars. The correctness of the retrieval code and accuracy of the proposed algorithm are validated using datasets generated by a numerical model under the OSSE framework. The retrieval scheme is applied using data collected during IOP 8 of the 2008 SoWMEX to retrieve the thermodynamic fields of a prefrontal squall line after it has made landfall in southern Taiwan, and reasonable results are obtained.

The terrain-resolving capability of WISSDOM and the proposed TPTRS introduced in this study are able to provide three-dimensional high-resolution pressure and temperature along with the wind fields over complex terrain, allowing us to obtain better diagnoses of the thermodynamic and kinematic structure of a heavy rainfall system, especially in mountainous areas. These observed and retrieved meteorological fields are also candidates ready to be assimilated into a numerical weather prediction model to improve its forecast skill.

Acknowledgments

This research is supported by the Ministry of Science and Technology of Taiwan under MOST107-2111-M-008-040, MOST107-2625-M-008-008, and by the Central Weather Bureau (CWB) under MOTC-CWB-107-M-02. The authors thank CWB for providing the radar and surface station data.

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