Some of the characteristics that the present authors explored here have also been discussed in previous studies, such as Hoskins and Hodges (2019). However, Hoskins and Hodges (2019) examined STs vertical changes for only two levels, 250 hPa and 850 hPa. Here our focus is to expand on the work of Hoskins and Hodges (2019) by analysing more levels in the vertical, including the lower stratosphere at 100 hPa, using a cyclone tracking algorithm. Hoskins and Hodges (2019) noticed a circumpolar ST in the NH at 250 hPa after observing the separation of Pacific and Atlantic STs at 850 hPa. Furthermore, STs are found further north in the boreal summer than in the winter, and STs at 250 hPa are stronger during boreal winter. Similarly, we also found these features in the present study. In the lower troposphere, even at 1000 hPa, the STs over the Atlantic and Pacific are separated because of the local strong baroclinic instability generation features such as the Canadian High and Siberian High generating high northern winds meeting northernward moving very warm southerly wind on the eastward of the Gulfstream and Kuroshio warm currents (Blackmon 1977). Hoskins and Hodges mention some characteristics of STs (2019). There are some other characteristics of STs that can be attributed to WDs that have not been discussed by prior studies. Here we discuss the WDs phenomenon in very detail and also discuss the vertical variations of the strength of the ST, extension, and seasonal variations in the following section of Results and Discussion.

This study is unique in that it discusses the vertical variations of ST characteristics in the vertical from troposphere to the lower stratosphere (100 hPa) during all boreal seasons, as shown in Table 1. Thus, a close examination of Table 1 allows one to get a concise idea of these variations. For example, WDs are not found at mean sea level pressure (MSLP) or 1000 hPa level and are obviously generated at higher levels only, but of course, they are strongest at 500 hPa during winter. A novel observation we found of STs is that they exist in the lower stratosphere, albeit weak. There was no comparison of WD in STs in earlier studies. The WDs are also generated by the baroclinic instability, as first found by Rao and Rao (1971) more than 50 years ago, like the middle and high latitude extratropical cyclones in STs, which are seen in other regions in the Atlantic and Pacific.

Table 1

The spatial extention (in °)of STs at MSLP, 1000 hPa, 800 hPa, 500 hPa, 300 hPa, 100 hPa.

Season | Winter | Spring | Summer | Autumn |

**Height** | **WD** | **Pacific** | **Atlantic** | **WD** | **Pacific** | **Atlantic** | **WD** | **Pacific** | **Atlantic** | **WD** | **Pacific** | **Atlantic** |

MSLP | - | 20–30 | 30°-40° | - | 10°-20° | 10°-20° | - | 10°-20° | 20°-30° | - | 10°-20° 30°-40° (Russia) | 20°-30° 30°-40° (USA) |

1000 hPa | - | 50°-60° | 60°-70° | - | 40°-50° | 50°-60° | - | 30°-40° | 50°-60° | - | 50°-60° (East) | 40°-50° (West) |

800 hPa | Middle-East 40°-50° N | 50°-60° (extended) | 50°-60° (small) | Middle-East 30°-40° | Horse Shoe 30°-40° (West) | Mid Atlantic 30°-40° | - | West 30°-40° | Mid Atlantic 30°-40° | - | West Pacific 50°-60° 40°-50° (Russia) | Mid Atlantic 40°-50° |

500 hPa | Strongest 60°-70° N Middle east 50°-60° N | Extended 50°-60° | Mid Atlantic 50°-60° | 50°-60° short | 50°-60° | 40°-50° | - | 40°-50° | 40°-50° 50°-60° (short) | Further North 40°-50° short center | 50°-60° | West 50°-60° |

300 hPa with continuous contours over 90° E & 150° E | Middle east 50°-60° N | 50°-60° | 50°-60° | Middle East 40°-50°, over 90°E 50°-60° | 150°E, 50°-60° | Weak 40°-50° | - | Weak 40°-50° | 50°-60° | - | 50°-60° | 40°-50° |

100 hPa weak around 30°N from 30°W to 90°E | 10°-20° N | Around 30°N from 180°E to East upto 120°W | At high latitudes, ST north of 60°N from 90°W to 90°E 10–20 | - | North of 30°N from 90°E to 150°W | Around 30°N Mid Atlantic 10–20 | - | North of 30°N from 30°E to 150°W, maximum Russia 20°-30° | Around 60°N shorter than spring season 10°-20° | - | Continuous ST, North of 30°N upto 0° Maxmimum over 180° 20°-30° | - |

In an detailed series of articles, Hunt et al. (2018a,b; 2019a,b) and Dimri et al. (2015) made an investigation of WDs. These are needed before any attempt is made to explain the nature of WD theoretically. Hunt et al. (2018a) show a maximum number of STs concentrated in the northwestern parts of India and Pakistan regions, which are the continuation of STs from the western Pacific region - Middle East - Northwestern parts of India and triggering heavy precipitation. This implies that there is a strong local forcing to raise the baroclinic instability (Figs. 2a,b). This needs a detailed study to emphasize the characteristics of the concentrated ST region of Pakistan and India. Here our primary focus is to discuss the ST and their WD association.

In fact, an accomplished atmospheric and oceanic scientist Prof. A. D. Gill, suggests a three-pronged approach to study and predict weather and climate, namely K-Knowledge (observational studies), U- Understanding (to find causal mechanisms of the determined effects theoretically) and finally M- Modeling (approach to deterministic prediction). In the case of WD, mainly K and a little M seem to be done by several researchers. But, U is almost lacking which needs to be focused with the help of observations. In our knowledge the only attemt seemed to be that of Rao and Rao (1971). Here we extend that simple study to explain the physical importance of STs and WDs. The recent observational study by Hunt et al. (2018a, 2018b, 2019a, 2019b, and 2021) found a westward tilt with height, which implies a northward transport of sensible heat (V’T’ > 0) as explained below. They seem to have neglected this important implification.

Assume a perturbation in the stream function Ψ (primes are omitted for simplicity )

$$\phi =A\left(P,t\right)sinkx+B\left(P,t\right)coskx$$

\({R}_{\phi }\left(P,t\right)sin{\left(kx+\delta \left(P\right)\right)}_{y}\) ; \(\vartheta =\frac{\partial \phi }{\partial x}=Rcos\left(kx+\delta \right)\) ---- (1)

$$\frac{\partial \varnothing }{\partial P}=f\frac{\partial \phi }{\partial P}=f\frac{\partial R}{\partial P}sin\left(kx+\delta \right)+fRcos\left(kx+\delta \right)\frac{\partial \delta }{\partial P}$$

So,\(\vartheta \frac{\partial \varnothing }{\partial P}={R}^{2}fk{cos}^{2}kx+\delta \frac{\partial \delta }{\partial P}+Rf\frac{\partial R}{\partial P}sin\left(kx+\delta \right)cos\left(kx+\delta \right)\)

Taking zonal average

\(\vartheta \frac{\partial \varnothing }{\partial P}=\frac{f}{k}\stackrel{`}{{\vartheta }^{2}}\frac{\partial \delta }{\partial P}\) since \(\vartheta =Rcos\left(kx+\delta \right)\)

Or \(\vartheta \left(\frac{-RT}{P}\right)=\frac{f}{k}\stackrel{`}{{\vartheta }^{2}}\frac{\partial \delta }{\partial P}\) as\(\left\{\frac{\partial \varnothing }{\partial P}=-Z=\frac{-RT}{P}\right\}\)

Or \(\stackrel{`}{\vartheta T}=\frac{-Pf}{R}\stackrel{`}{{\vartheta }^{2}}\frac{\partial \delta }{\partial P}\) --- (2)

If the trough or ridge tilt westward with height as observed by the Hunt et al. (2018a) \(\frac{\partial \delta }{\partial P}<0\), \(\stackrel{`}{\vartheta T}>0\) or northward transport of the sensible heat.

Also, for a simple quasi-geostrophic atmosphere, the energy exchanges can be derived (see Holton 2014 for more details) as

\(\frac{ \partial K}{\partial t}=C\left(P,K\right)\) --- (3)

\(\frac{\partial P}{\partial t}=C\left({P}_{z},P\right)-C\left(P,K\right)\) --- (4)

Where C(P, K) is the rate of conversion between the available potential energy of the eddy and eddy kinetic energy given by

\(C\left(P,K\right)=\int \stackrel{`}{\omega T}dm\) --- (5)

\(C\left({P}_{z},P\right)=-\int \frac{R}{P\sigma }\frac{\partial U}{\partial P}\stackrel{`}{\vartheta T}dm\) --- (6)

From Eq. (6), we note that if there is a northward transport of sensible heat by WD (\(\stackrel{`}{\vartheta T}>0\)) for vertically increasing subtropical westerlies i.e. i.e., \(\frac{\partial U}{\partial P}<0\), then \(C\left({P}_{z},P\right)>0\). This means that mean available potential energy gets converted into perturbation (for WDs) kinetic energy.

Also, for warm air rising and cold air sinking zonally, \(C\left(P,K\right)>0\).

That is, the energy cycle of WD would be:

\({P}_{z}\to P\to K\) -- (7)

Equation (7) is the typically unstable perturbations energy cycle found for WD by Rao and Rao (1971). In the case of WD, Hunt et al. (2018a) found a westerly tilt with a height that is \({P}_{z}‚ÜíP\)occurs. But, in their Figures of anomalies Fig. 4, the waves with warm air rising and cold air sinking are not clear. If they had shown actual temperature values instead of anomalies, they might have seen warm air raising in front of the WD and cold air sinking behind the WD trough.

The perturbation quasi-geostrophic omega equation can be derived as

\(\frac{{\partial }^{2}\omega }{\partial {x}^{2}}+\frac{{f}_{o}^{2}}{\sigma }\frac{{\partial }^{2}\omega }{\partial {P}^{2}}=\frac{f}{\sigma }\left[2\frac{\partial U}{\partial P}\frac{\partial }{\partial x}\left(\frac{{\partial }^{2}\psi }{\partial {x}^{2}}\right)+\beta \frac{\partial \vartheta }{\partial P}\right]\) --- (8)

Where symbols have usual meaning. Then if we make the following simplification

\(\frac{{\partial }^{2}}{\partial x}=\frac{-1}{{L}^{2}}\) ; \(\frac{{\partial }^{2}}{\partial x}=\frac{-1}{{P}^{2}}\) --- (9)

Also, for a simple quasi-geostrophic atmosphere

$$\omega =\frac{f{p}^{2}}{\sigma {p}^{2}+{L}^{2}{p}^{2}}\left[2\frac{\partial U}{\partial P}\vartheta -\beta {L}^{2}\frac{\partial \vartheta }{\partial P}\right]$$

For waves of wavelength for WD ( small scale waves ), the second term, the *beta* in the bracket, can be neglected because L square is small, Then

$$\omega =\frac{f{p}^{2}}{\sigma {p}^{2}+{L}^{2}{p}^{2}}\left[2\frac{\partial U}{\partial P}\vartheta \right]$$

Ahead of WD \(\vartheta\)>0 and \(\frac{\partial U}{\partial P}\) is negative for the subtropical jet with strong westerlies and vertical shear i.e.,\(œâ\)<0. On the other hand, ahead of the trough, there would be a raising motion, as observed by Hunt et al. (2018a). This shows that WD has all the typical characteristics of baroclinically unstable wave disturbances, as earlier discussed by Rao and Rao (1971).