Starting from the vector relationship of three momentum equations for frictionless flow ( and the differential form of the definition of equivalent potential temperature the following equation is gained 6 where the specific entropy ( and enthalpy ( are introduced. The absolute velocity vector with , , and denoting the relative velocity vector, the angular velocity of rotating frame, and the radius of the earth; θ, q_s, T, , and L represent the potential temperature, saturated specific humidity, absolute temperature, the specific heat of dry air at constant pressure p, and the latent heat of vaporization, respectively.

In Cartesian coordinates, the momentum vector, Eq. (6), expressed in terms of the absolute velocity components, can be written in components as 7 In Lagrangian coordinates, equation (7) is rewritten as 8 where the subscript L denotes the Lagrangian system. We use the vector to denote the Lagrangian particle labels or “enumeration coordinates”, which we define as the coordinates of the spatial position of the fluid blob at t=0. There is a one-to-one map between the enumeration coordinates and the vector , which gives the position of the particle at time t whose initial position is . We assume that both and its inverse are differentiable everywhere. By multiplying both sides of Eq. (8) by and then summing over k, we obtain 9 Equation (9) can be rewritten as 10 with For brevity, is defined as the Lagrangian function in the absolute reference frame.

To derive an auxiliary relation needed to justify Eq. (11) below, we note that at the initial time (t = 0), the Lagrangian and Eulerian systems coincide so that . This implies that . This is equal to the usual Kronecker delta function , which is defined as one when its two subscripts, i and n, are equal and zero otherwise. Therefore,

After equation (10) is integrated with respect to time for moist isentropic flow ( , we obtain the following expression: 11 where and are the action integrals of the absolute reference system. By their definition as particle labels, defined as the initial positions of the particles, the enumeration coordinates a do not change with time ( .

Returning to the xⁱ system by a series of simplification similar to ZB03, then there will be12 The above equations are multiplied by the unit vectors , , for k = 1,2,3, respectively. By adding the results and using the definitions and with , and , we obtain the absolute velocity 13 The suffix 0 refers to t = 0, and the notation .

From Eq. (13), move the terms and on the right-hand side to the left-hand side of Eq. (13), and define 14a which implies 14b then we will obtain the expression , here is dubbed the “baroclinic ERI” in moist flow.

Equation (13) reminds us to recall some basic results under the Clebsch transformation.^[7–9] According to the Clebsch transformation^[7] for the velocity field in barotropic flow ( and its generalization in non-barotropic flow proposed by Herivel^[8] ( , Seliger and Whitham^[9] obtained two additional potentials expressed as 15 where corresponds with the first three terms on the rhs of Eq. (13), associated with the initial velocity, and and correspond to the last two terms on the right-hand side of Eq. (13). Thus, formula (13) coincides formally with Eq. (15).

Seliger and Whitham^[9] indicated that the velocity field could be written as Eq. (15) for the most general flow. Equation (13) derived above is the further generalization of Eq. (15) to moist flow.

Compared with our previous study,^[12] the present study gives a simple physical explanation of baroclinic ERI in moist flow on the basis of the more in-depth understanding of the formulism. From Eqs. (14a) and (14b) and the definition of the ERI in moist flow in section 2, the expression of is deployed as the dot product between and . Like the previous name,^{[6, 11, 17]} and are called the generalized velocity and generalized vorticity in moist flow, respectively. Both in and in represent the baroclinic effect, which exactly constitutes the difference between the barotropic^[5] and baroclinic ERI expressions.^[11] Based on the concepts of generalized velocity and generalized vorticity, is the generalized helicity in moist baroclinic atmosphere. Therefore, by introducing moist enthalpy (h_m in and moisture entropy ( in the dry ERI formula ( , the moist ERI is derived, and the moist ERI coincides formally with the dry version.

Thus, we have answered questions 1 and 2 mentioned in the introduction: we can further extend the derivation of the continuity equation in general coordinate to moist flow with mass source/sink term, and the quantity of the so-called ERI does keep its usual expression in moist fluid with mass forcing. That is to say, the moist ERI formula ( derived here coincides formally with the dry version ( ),^{[5, 11]} except that specific enthalpy (h in and specific entropy (s) are replaced by moist enthalpy (h_m in and moisture entropy ( in the present study by considering the moisture effect. To answer question 3, we now prove whether the baroclinic ERI in moist flow remains conserved following the flow given mass forcing.

It seemed that the former two problems aforementioned in the introduction have been solved. A small question should be noted. At a quick glance over the above derivation, whether the mass forcing is ignored or not, the ERI expression keeps its usual form because the continuity equation derived in section 2 is not utilized therein during the overall derivation in section 3. However, mass forcing makes ERI no longer conserved (the detailed proof will be given in the next subsection). Thus, from the definition itself, the so-called ERI is no longer an invariant because it is conditionally conserved. Maybe renewing the definition and name is a more appropriate choice. Thus, it should be renamed a quasi-invariant on the premise of conditional conserved. But we still follow the utility of ERI below in order to respect many previous researchers (e.g., Ertel and Rossby;^[5] Zdunkowski and Bott^[11]).

Due to the interaction among several typhoons, multi-typhoon systems are complicated. The tracks and landing points are more difficult to capture. A triple-typhoon system comprised of Typhoons “Lionrock”, “Kompasu”, and “Namtheun” which attacked China together in 2010 is studied here. “Lionrock” began to be named and numbered at 06 UTC on 28 August 2010, formed as a tropical cyclone above the South China Sea at 06 UTC on 28 August 2010, then moved northwards and intensified. After a “question mark”-form path, it landed at ZhangPu county in Fujian province with a maximum wind speed of and a minimum pressure of 990 hPa at about 23 UTC on 1 September 2010. It weakened gradually during its westward movement, but had a long lifetime, lasting about 6 days as an identifiable tropical storm. Compared with “Lionrock”, Typhoon “Kompasu” was a relatively fast-moving typhoon that traveled far during its lifetime of only four days. Typhoon “Namtheun”’ lived less than 2 days. It had a track that was shorter than “Kompasu” and not so complicated as “Lionrock”. The long-term stagnation of “Lionrock” and the long-range travel of “Kompasu” brought especially long-lasting, disastrously heavy precipitation over a wide area, which caused enormous casualties and economic loss.

To resolve tropical cyclones and be suitable for a fast manifold analysis, a 3D non-hydrostatic weather research and forecast (WRF) model (V3.7.1) is used to generate a set of meso-scale meteorological fields for the fast manifold analysis. The detailed parameterization scheme and microphysical process are similar to those in previous studies.^{[20, 21]} The WRF is configured covering the wide area (111° E–145° E, 10° N–50° N) with a horizontal resolution of 3 km. For the model domain, 6 separate WRF simulations are performed with each one starting at 00 UTC and lasting 30 hours. The first 6 h of each run is discarded as a spin-up run, i.e., each simulation overlaps the previous one to account for model spin-up time. Thus a set of mesoscale meteorological databases with 1-hour interval model outputs is obtained, spanning from 06 UTC 28 August to 06 UTC 3 September 2010. Note that, 6 separate WRF simulations are performed and each simulation overlaps the previous one herein. This design scheme is to avoid the larger and larger simulation errors producing with time, so that the simulated results do not depart from the observation too much.

The comparison between the WRF simulations and the observations is performed firstly. Figure 1 shows the comparisons of the sea-level pressure and the maximal wind speed between simulations and observations for three typhoons during their individual lifetimes. The intensity and evolution of sea-level pressure of “Namtheun” (green solid line in Fig. 1(a)) are simulated better than those of “Lionrock” and “Kompasu” because of its short lifetime and simple track. Due to complex track and long lifetime of the “Lionrock”, both the observed and the simulated sea-level pressure evolutions experience several fluctuations accompanied with the increase or decrease of typhoon intensity (blue solid line in Fig. 1(a)). They have a maximum derivation (no more than 10 hPa), and the simulation sea-level pressure is slightly smoother than the observed one. Generally speaking, both the observation and simulation have the synchronous increase or decrease trend with time. For typhoon “Kompasu”, the simulated sea-level pressure evolves also consistently with the observed one, descends then increases during its lifetime (orange solid line in Fig. 1(a)). The simulated maximal wind speeds for three typhoons experience basically consistent evolutions with the observations (Fig. 1(b)). There are not too large gaps between the simulation and the observation. Like the sea-level pressure shown in Fig. 1(a), the maximal wind speed for “Lionrock” also has several waves accompanied with the fluctuations of typhoon intensity.

Fig. 1.

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	Fig. 1. (color online) Observed (solid line) and simulated (solid line with different symbols marked) evolutions of (a) sea-level pressure (hPa) and (b) maximal wind speed ( for three typhoon “Lionrock” (blue), “Namtheun” (green), and “Kompasu” (orange).

Figure 2 shows the simulated tracks (black solid line with dots) of three Typhoons “Lionrock”, “Namtheun”, and “Kompasu” from 06 UTC 28 August to 18 UTC 2 September 2010. The simulated typhoons paths of “Namtheun”, and “Kompasu” are very close to the observations (http://typhoon.zjwater.gov.cn/default.aspx). Even for the complex moving path of Typhoons “Lionrock”, the simulation does not depart from the observation too far. The purpose of this simulation is to create an in-physics consistent dataset with what would be needed to resolve tropical cyclone and to be suitable for a fast manifold analysis. The simulation result presented above should basically satisfy our main objective.

Fig. 2.

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	Fig. 2. (color online) Tracks of moist ERI and MPV anomaly centers (red and green dash lines with triangles), superposed with simulated (black solid line with dots) tracks of three Typhoons “Lionrock”, “Namtheun”, and “Kompasu” from 06 UTC 28 August to 18 UTC 2 September 2010.

Using the simulation data, the evolutions of the moist ERI anomaly and the MPV anomaly are analyzed for this triple-typhoon case and their landing on the Chinese main land. At 700 hPa, the coherences between the cyclone center and the strong signal of moist ERI anomaly are clearly shown in Figs. 3(a) and 3(b). Note that the moist ERI anomalies shown in Fig. 3 might be produced by the source/sink term in Eq. (22), which may be a possible application of ERI nonconservation to real moist atmosphere for fast-manifold flow. At 12 UTC 30 August (Fig. 3(a)), “Namtheun” was located in the north of Taiwan island at (122.6° E, 26.1° N) with minimum pressure 996 hPa and maximum wind speed , “Lionrock” was located at (116.9° E, 21.1° N) with an intensity of (980 hPa, , and “Kompasu” was located at (131.4° E, 23.8° N) with a stronger intensity of (975 hPa, . The three typhoon locations determine a triangle distributed in the south of 30° N along the southeast coastline of China. The centers of moist ERI anomalies (shaded) are superposed with the centers of three typhoons with stronger signals in “Kompasu” than in “Lionrock” and “Namtheun”. After 18 UTC 31 August 2010 (Fig. 3(b)), Typhoon “Namtheun” disappeared gradually and the triple-typhoon system tended to evolve into a binary-typhoon (BT) system comprised of Typhoons “Lionrock” and “Kompasu”. The moist ERI anomalies evolved correspondingly. Firstly, the centers of three moist ERI anomalies became two and moved following the BT system. Secondly, both their intensities identically varied. For example, the moist ERI anomaly was strengthened following the intensification of Typhoon “Kompasu” (965 hPa, , and the moist ERI anomaly was weakened following the weakening Typhoon “Lionrock” (990 hPa, . At the same time, the strongest Typhoon “Kompasu” still corresponded with the largest moist ERI anomaly.

Fig. 3.

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	Fig. 3. (color online) Horizontal distributions of (a) moist ERI anomaly (shaded, at 12 UTC 30, (b) moist ERI anomaly at 18 UTC 31, (c) MPV anomaly (shaded, ) at 12 UTC 30, and (d) MPV anomaly at 18 UTC 31 August 2010; superposed with geopotential height (isolines, dgpm) at 700 hPa.

For comparison, the MPV anomalies at 700 hPa are also computed (Figs. 3(c)–3(d)). In contrast with the moist ERI at 12 UTC 30 August (Fig. 3(a)), which was coherent and well-correlated with the typhoons, the MPV distributions for the triple-typhoon system were scattered (Fig. 3(c)). Their maximum values were located in the north of 40° N. After 18 UTC 31 August, MPV anomalies did not concentrate on typhoons (Fig. 3(d)). The contrasting behaviors of moist ERI and MPV at 700 hPa indicate that the moist ERI in lower troposphere is a better diagnostic tool for typhoons because it concentrates around the centers of typhoons and can show the relative intensity variations of several coexisting typhoons. Moist ERI anomaly can track the movement and development of multi-typhoon systems as well as individual storms.

Returning to Fig. 2, the tracks of the moist ERI and MPV anomaly centers (red and green dashed lines with triangles) are depicted, superposed with simulated (black solid line with dots) tracks of three typhoons during their lifetimes (“Lionrock” from 06 UTC 28 August to 18 UTC 2 September, “Namtheun” from 06 UTC 30 to 18 UTC 31 August, “Kompasu” from 06 UTC 29 August to 06 UTC 2 September, 2010). It shows that the tracks of moist ERI anomaly centers are similar to those of the simulated typhoon centers. Although the MPV anomalies only in the vicinity of centers of typhoons are considered (cf. the large MPV anomaly in the high-latitude region is excluded because it is far from typhoons), ERI anomaly centers at many times are located between the MPV and the simulated typhoon centers. That is to say, the tracks of two equivalences, between ERI and MPV anomalies, are compared. The better collocation between simulated track and ERI anomaly center than that between simulated track and MPV anomaly does mean an improvement. The moist ERI formula focusing on the fast-manifold effect might explain this phenomenon. The moist ERI anomaly centers always are at or ahead of the typhoon centers except that there is a little deviation at the beginning and also after landfall. Even for the fast, long-ranging Typhoon “Kompasu” and the complicated path of Typhoon “Lionrock”, the moist ERI anomaly centers still approach to the typhoon centers, and the landfall positions of the moist ERI anomalies and the corresponding typhoons are close. So the moist ERI could be used as a tracer of both the typhoons tracks and their changing intensities.