Indicators for the Compression and Stretching Characteristics of the HTF-Coordinate of WRF

The HTF-coordinate a feature the Weather and Forecasting model and it is importanttounderstandhowit canbeused eﬃciently.Indicators areneededtodescribethecharacteristicsof thedistribution ofthevertical coordinate and to evaluate the rationality of the conﬁguration of the HTF-coordinate. Such indicators, including the maximum factor of compression (MFC), eta corresponding to the MFC, and the critical point of compression (CPC), were deﬁned and calculatedin this study. The indicatorswere alsovalidated toeﬃciently andaccurately describethecharacteristicsof the distributionof the vertical coordinate of the WRF model. They constituted the integrated system to resolve the problem of the conﬁguration of the HTF-coordinate of WRF. The nonmonotonicity, compression, and stretching that were caused by the nonlinearity of the HTF-coordinate were also illustrated and validated.


Introduction
e terrain-following coordinate (TF-coordinate), which promotes the effective application of numerical weather prediction (NWP) models, has been used widely in various dynamic frameworks of such models [1][2][3][4]. However, it has certain defects, e.g., the use of terrain-following coordinates reduces the accuracy of the horizontal pressure gradient force (PGF) in regions of relative steep terrain [5] and divergent wind anomalies extend from the bottom to the top levels of the model and distort the terrain gravitational wave in steep terrain [6]. In high-resolution models, as the slope of the steep terrain becomes larger, the spurious disturbance caused by the numerical errors of the horizontal PGF will become more distinct [7], weakening the prediction performance of the high-resolution numerical model. To help mitigate these effects, various modifications to the TF-coordinate formulation have been introduced that more rapidly remove terrain influences in the coordinate surfaces with increasing height. e most successful and widely used approach is the hybrid terrain-following coordinate (HTFcoordinate) [8][9][10].
e HTF-coordinate smoothly transitions from the TF-coordinate at the surface to the purely isobaric coordinate (p-coordinate) at the higher levels of the model to diminish the effect of steep terrain [11][12][13][14][15][16][17]. rough the implementation of the hybrid coordinate, the WRF model significantly reduces small-scale spurious vertical velocities, particularly at upper levels and downstream of complex terrain [18,19].
WRF is a regional numerical weather prediction model developed by the National Center of Atmospheric Research (NCAR) and is used worldwide. WRF v4.0 implements NCAR's HTF-coordinate using the Klemp cubic polynomial approach [20]. How to configure the parameters and take full advantage of the HTF-coordinate are important in the application of the hybrid coordinate system. Park et al. [21] deduced the necessary condition to maintain the monotonicity of the HTF-coordinate, which is an important result for the operational application of the HTF-coordinate. Indicators are also needed to describe the characteristics of the distribution of the vertical coordinate to choose the rational parameters of the HTF-coordinate of WRF. In this paper, the nonmonotonicity, compression, and stretching of the vertical coordinate were studied and the indicators of compression and stretching were defined, calculated, and validated. ey constituted the indicator system to resolve the problem of the configuration of the HTF-coordinate.
Our paper is organized as follows. In Section 2, the basic contents and characteristics of the HTF-coordinate are introduced briefly. In Section 3, the nonmonotonicity of the HTF-coordinate is introduced and illustrated using idealized tests. In Section 4, the difference of the partial derivative of normalized pd with respect to eta, the MFC, eta corresponding to the MFC, and the CPC of the HTF-coordinate are defined and calculated. ey are then validated to allow the effective and accurate evaluations of the compression and stretching of the vertical coordinate in an idealized experiment. Section 5 presents some discussion. In Section 6, the conclusions of the study are summarized.

HTF-Coordinate Introduction
is is a brief introduction to the basic contents of the HTFcoordinate to facilitate the calculations in the below sections.
2.1. TF-Coordinate. TF-coordinates are used in mesoscale numerical weather prediction models to simplify the processing of surface boundary conditions. e TF-coordinate in WRF has the form where p d (noted as pd) is the hydrostatic component of the pressure of dry air and p t (noted as pt) and p s (noted as ps) are the hydrostatic top level pressure and the surface pressure for dry air, respectively. μ is the difference of hydrostatic pressure of dry air from the top level to the surface of the model; subscript d indicates dry air. η (noted as eta) indicates the vertical coordinate value of the model [18]. When p d � p s and η � 1, the vertical coordinate value is always 1 at the surface boundary.

HTF-Coordinate.
e WRF HTF-coordinate is given as where p 0 is the reference sea level pressure of air, B(η) defines the relative weighting between the TF-coordinate and a purely isobaric coordinate, and B(η) smoothly transitions from the TF-coordinate at the surface to the purely isobaric coordinate in the upper levels of the model [18][19][20][21]. B(η) is defined in terms of a third-order polynomial: Subject to the boundary conditions of equation (4), where η c (noted as eta_c) is a user-defined constant that specifies where the vertical coordinate completely transitions from the TF-coordinate levels at low levels to purely isobaric coordinate levels aloft. e subscript η represents the partial derivative with respect to eta. When η � 1, the model applies the TF-coordinate at the surface; when 0 ≤ η ≤ η c , the model applies the p-coordinate in the upper levels, and when η c ≤ η ≤ 1, the transitions from TF-coordinate to p-coordinate take place.

The Nonmonotonicity of the HTF-Coordinate
Although the HTF-coordinate reduces the errors in the PGF, it also generates other defects. For example, it cannot maintain the monotonic, continuous, and smooth one-toone mapping relationship between eta and p d at the lower levels of the model when eta_c is set to excessively large values in regions of steep terrain, which causes the adjacent coordinate surfaces to intersect with each other, violating the monotonicity requirement of the coordinate.  Table 1. Four test cases were applied. From Figure 1, the following conclusions can be drawn:

Advances in Meteorology
Experiment 2. To further analyze the mapping relationship between eta and pd, an experiment with an ideal terrain was designed. e terrain was designed as a sine function: where A is the amplitude of the sine function and p s represents the height of the terrain. e parameters for Experiment 2 are listed in Table 2.
From Figure 2, the following conclusions can be drawn:  e maximum value of eta_c should be studied to simulate WRF. To maintain the monotonic mapping relation between eta and pd, Park [21] found that p s and eta_c should conform to the following equations: where p s, min is the smallest allowable surface pressure for which the hybrid coordinate does not violate monotonicity. p * s, min is the normalized p s, min , p * s is the normalized hydrostatic surface pressure of the model, and p * t is the normalized pressure at the top level of the model [21].
However, to ensure that the solution remains accurate and stable, it is important to configure the hybrid coordinate in a way that the anticipated minimum surface pressure p * s is significantly greater than p * s, min . According to (7), we created a relational table of p * t , p * s, min , and eta_c as shown in Table 3.

Compression and Stretching Experiment (Experiment 3).
An experiment was designed and conducted to illustrate the compression and stretching of the vertical coordinate caused by the HTF-coordinate. e vertical coordinate of the model was divided into 50 even levels between 0 and 1. e configuration of Experiment 3 is listed in Table 4.
As shown in Figure 3, pd was distributed evenly by the eta value in the case of the TF-coordinate. With the same configuration, the distribution of pd was more concentrated at the lower levels and more decentralized at the higher levels of the model in Figures 3(b)-3(d). When eta_c was larger, the compression and stretching was more significant. At high levels of the model, the HTFcoordinate transitioned into a purely isobaric coordinate when η ≤ η c , which caused pd to change almost linearly with eta.

Estimating the Compression and Stretching of the Vertical
Coordinate.
e compression and stretching of the vertical coordinate can be estimated using the difference of the partial derivative of the normalized pd with respect to eta of the HTF-coordinate and that of the TF-coordinate. e rationality of the approach lies in the value of the partial derivative of the normalized pd with respect to eta of the TF-coordinate being a constant as the pt * is constant that can be standardized for the comparison. e difference represents the relative rate of change of normalized pd with respect to eta between the TF-and the HTF-coordinate. If the difference is negative, this indicates compression of the vertical coordinate, and if the difference is positive, this indicates stretching. e greater the positive difference, the more significant the stretching. Conversely, the smaller the negative difference, the more significant the compression. e equation is derived as follows: Equation (8) is the expression of the partial derivative of the normalized pd with respect to eta of the TF-coordinate, and equation (9) is the expression of the HTF-coordinate. Equation (10) presents the difference of the partial derivative of the normalized pd with respect to eta between the  (9) and (10) are piecewise functions because the HTF-coordinate completely transitions into the purely isobaric coordinate when η ∈ (0, η c ). From Figure 4, the following conclusions can be drawn:

MFC of the HTF-Coordinate.
It is reasonable to estimate the compression using the minimum difference of the partial derivative of the normalized pd with respect to eta. e MFC is defined in terms of the absolute value of the minimum difference and is calculated as follows: η c ∈ (0, 1), p * s ∈ (0, 1).

(12)
As shown in equation (12), the MFC a positive value. e compression of the vertical coordinate is more significant with an increasing MFC. Note that eta_c should not be set to zero as this would result in the MFC being infinity.
MFC is only relevant to the normalized surface pressure and eta_c. From Figure 5, the following conclusions can be drawn: (iii) MFC is zero when the normalized surface pressure is set to 1, which means that no compression or stretching of the vertical coordinate occurs.
e MFC of the vertical coordinate was calculated based on equation (12) as shown in Table 5.

Eta
Corresponding to the MFC. Eta corresponding to the MFC indicates eta where the compression of the vertical coordinate is most significant and is calculated using the following equation: Eta is only relevant to eta_c. e relationship between eta corresponding to the MFC and eta_c is shown in Figure 6.
Based on Figure 6, eta corresponding to the MFC increases with increasing eta_c and is always less than eta_c. e values of eta corresponding to the MFC were calculated and are listed in Table 6.

CPC of the HTF-Coordinate.
e vertical coordinate of the HTF-coordinate is divided into two intervals. One   To be able to conveniently look up the critical point, the CPC is listed in Table 7.
As shown in Figure 7, the vertical coordinate is compressed in the region above the curve and stretched in the region under the curve. When eta_c increases, the compressed region gradually decreases and the stretched region gradually increases accordingly.

Validation of the Compression of HTF-Coordinate eory.
e compression theory was validated using parameters from Experiment 3, where the normalized surface pressure is 0.6 and eta_c is set as 0.0, 0.2, and 0.4, respectively. e MFC, eta corresponding to the MFC, and the CPC are calculated and listed in Table 8.
Based on Table 8 and Figure 3, the MFC is 0.5143 when eta_c = 0.4, which is reasonable because the compression of the vertical coordinate is most significant in Figure 3 e CPC is 0.4857 when eta_c = 0.4, which indicates the location of the completely transition in Figure 3(d). According to the validation, the compression theory of the HTF-coordinate can be used to efficiently estimate the compression and stretching of the vertical coordinate of the HTF-coordinate.

Discussion
Although WRF is used widely and the HTF-coordinate is imported into WRF v4.0, the accurate configuration of the hybrid coordinate is not explicitly noted in the WRF guide. Park et al. [21] systematically studied the nonlinearity of the HTF-coordinate of WRF in 2019 and stated the necessary conditions to maintain the linearity of the HTF-coordinate, which is important for the application of the HTF-coordinate. However, more indicators are needed to reveal the characteristics of the distribution of the vertical coordinate of the HTF-coordinate and to understand where the vertical coordinate is mainly compressed, where the compression transitions to stretching, and the degree of the compression that is important for the operational application of WRF. e difference of the partial derivative of the normalized pd with respect to eta, the MFC, eta corresponding to the MFC, and the CPC of the HTF-coordinate are defined, calculated, and validated to evaluate the compression and stretching of the vertical coordinate effectively and accurately. However, further analysis is needed to understand which distribution of the vertical coordinate is best suited for WRF to reduce the numerical errors of PGF over steep terrains.

Conclusions
(i) e nonmonotonicity of the HTF-coordinate indicated that the mapping relationship between eta and pd was not monotonic, which was validated by the results of Experiments 1 and 2. is potentially caused the adjacent coordinate surfaces to intersect with each other, which violated the monotonicity

Data Availability
e data used to support the findings of this study are included within the article.