Thermal Wind Imbalance along the Curved Streamline of the Secondary Circulation in Tropical Cyclones

The controversy about the thermal wind balance (consisting of hydrostatic balance and gradient wind balance) at tropical cyclone (TC) eyewall has lasted for a long time. Several previous studies suggested that the tangential wind of TCs satisfied the state of thermal wind balance (La Seur and Hawkins, 1963; Hawkins and Rubsam, 1968; Jorgensen, 1984). Willoughby (1979) analyzed the scale of the cylindrical equation and obtained the hydrostatic balance and the gradient wind balance at zero order. Willoughby (1990) demonstrated that the gradient wind balance is effective in the free atmosphere around the centers of the Atlantic TCs by analyzing the aircraft observations. Based on thermal wind balance, some idealized TC theories were established (Charney and Eliassen, 1964; Emanuel, 1986; Wirth and Dunkerton, 2009). According to Eliassen (1951), the Sawyer–Eliassen equation based on the thermal wind balance is usually used to diagnose the secondary circulation of the TCs (Willoughby, 1979; Shapiro and Willoughby, 1982). However, several recent observational studies indicated that there is indeed a gradient wind imbalance in actual TCs (Schwendike and Kepert, 2008; Montgomery et al., 2014). Some numerical studies suggested that thermal wind balance in TC eyewalls might not be satisfied. Braun (2002) simulated Hurricane Bob (1991) and found that the buoyancy in the updraft of the eyewall is positive. Bryan and Rotunno (2009) simulated the TCs using an axisymmetric numerical model and found that the flow within eyewall regions is unbalanced. The gradient wind balance was also found to be disrupted at the top of simulated TCs (Cohen et al., 2017). For studies on boundary layer models, the super-gradient flow is a common feature of the output results (Kepert, 2001; Smith, 2003; Smith and Vogl, 2008), which is attributed to the advection of absolute angular momenta (Kepert, 2001). Wang et al. (2013) showed that there is a strong positive net radial force in the boundary layer during the second eyewall formation, which also supported the view that there are unbalanced dynamics in the boundary layer and that thermal wind balance cannot explain the second eyewall formation (Wang et al., 2016).

A compromise view is that thermal wind balance may be just an approximation, which has been widely recognized. However, there are still many ambiguities that are not conducive to ending the controversy. Bryan and Rotunno (2009) demonstrated the importance of unbalanced flow in the TCs. This study aims to clarify that the hydrostatic and gradient wind balance cannot be accurately satisfied simultaneously on the curved streamline of the TC secondary circulation.

The remainder of this paper is organized as follows. Section 2 describes the relationship between the accelerations (radial and vertical) and the curvature on streamline. In Section 3, the critical threshold of the curvature that allows thermal wind balance approximation is discussed based on the momentum equations. The main conclusions and discussion are presented in Section 4.

The curve movement has been discussed in previous studies. The relationship between curvature and acceleration has been proposed by Kreyszig (1991), as shown in Eq. (1),

where $ \kappa $ denotes the curvature of the trajectory, and $\boldsymbol{V}$ and ${\boldsymbol{V}}'$ indicate the vectors of velocity and acceleration. The trajectory of a steady-state flow coincides with the streamline.

Considering the air movement in the secondary circulation of an axisymmetric TC, the velocity vector ($ {\boldsymbol{V}}_{rz} $) consists of two components: radial velocity (u) and vertical velocity (w). The accelerations of the two velocity components are $ {\rm D}{u}/{\rm D}{t} $ and $ {\rm D}{w}/{\rm D}{t} $. Assuming that $ \kappa $ is positive (negative) when the center of curvature is located on the outside (inside) of the trajectory, Eq. (1) can be expressed as Eq. (2) below,

If the air moves along the curve, i.e., $ \kappa \ne 0 $ and $ \left|{\boldsymbol{V}}_{rz}\right|\ne 0 $, $ {\rm D}{u}/{\rm D}{t} $ and $ {\rm D}{w}/{\rm D}{t} $ cannot be zero simultaneously. This implies that thermal wind balance cannot be well satisfied on the curved streamline. The more curved the streamline is, the more obvious the thermal wind imbalance is.

So far, no streamlines of the secondary circulation in a steady TC have been captured in an actual situation. However, several common features of the streamlines can be identified in some simulation studies. For example, the azimuthally and temporally averaged radius–height structure of the simulated Hurricane Andrew (1992) indicated that the secondary circulation flows inwards at lower levels, turns upward sharply in the eyewall and expands outward at upper levels (Liu et al., 1999). A similar structure of the secondary circulation streamline is also shown in a TC analysis model (Bryan and Rotunno, 2009). Based on these common features, a conceptual model of the TC secondary circulation streamline is presented in Fig. 1.

Figure 1.  Streamline (solid curve) of the secondary circulation in TC eyewall on the radius–height (r–z) plane. The variable $ \beta $ denotes the angle between the moving direction (dashed) along the streamline and the vertical axis.

The variable $ \beta $ in Fig. 1 represents the angle between the moving direction along the streamline and the vertical axis of a TC, which satisfies the following relations,

where r and z indicate the coordinates in the radial and vertical directions, respectively. Equation (2) can be divided by w to obtain Eq. (5),

With Eq. (5), we can estimate the acceleration using streamline curvature.

For an approximately axisymmetric system, the radial and vertical momentum equations in cylindrical coordinates can be referred to in Smith (1980), as presented in Eqs. (6)–(7),

where $ v $ indicates the tangential wind of TCs, $ f $ is the Coriolis parameter, $ \mathrm{\alpha } $ is the specific volume, $ p $ is the pressure, $ g $ is the acceleration of gravity, and $ {F}_{u} $ and $ {F}_{w} $ represent the turbulent diffusion terms of u and w. Since the air movement above the boundary layer is approximately inviscid, the turbulent diffusion terms can be ignored. When the accelerations are small enough to be ignored, the thermal wind balance can be approximately satisfied.

According to the scales listed in Willoughby (1979), namely u = 10 m s⁻¹, w = 10 m s⁻¹, v = 50 m s⁻¹, r = 15 km, and z = 15 km, the centrifugal acceleration ($ -{v}^{2}/r $) is about 1.7 × 10⁻¹ m s⁻². At 20°N, the Coriolis parameter $ \left(f\right) $ is about 5 × 10⁻⁵ m s⁻¹. Thus, the Coriolis force ($ -fv $) is about 2.5 × 10⁻³ m s⁻¹, which is considerably smaller than the centrifugal acceleration in TC eyewall. Under the balance approximation, the magnitude of the centrifugal (gravitational) acceleration is the same as that of the radial (vertical) pressure gradient, both of which can be used as reference terms to identify whether the magnitude of the radial (vertical) acceleration term in TC eyewall is sufficiently small. We used centrifugal and gravitational accelerations as reference terms for the radial and vertical accelerations, respectively. The conditions satisfying the thermal wind balance approximation in TC eyewall can be expressed as Eqs. (8)–(9),

where ϵ indicates a critical coefficient. When discussing thermal wind imbalance, a 20% deviation from the pressure gradient was taken as an example in the previous study (Bryan and Rotunno, 2009). In this research, we also take 0.2 as the value of $ \epsilon $ for calculation.

For a mature TC, the structure is highly axisymmetric and close to the steady state. Ideally, the TC is assumed to be axisymmetric and in the steady state in this study. Therefore, the total derivative of w can be written as Eq. (10) by referring to Huang et al. (2019) as follows,

where $ \psi $ denotes constant variables or a conserved quantities along the streamline, i.e., the stream function, the absolute angular momentum, or the entropy in an adiabatic process. Hence, Eq. (10) expresses that the vertical acceleration is equal to the vertical variation of vertical kinetic energy along the streamline. Substituting Eq. (10) into Eq. (9), the condition of hydrostatic balance approximation can be expressed as Eq. (11),

Equation (11) indicates that when the airflow rises by 1 m along the streamline, the change of w² should be lower than 3.9 m s⁻¹. For the azimuthally averaged state in an actual TC eyewall, the variation of $ w $ is considerably lower than the critical value. Therefore, the hydrostatic balance approximation is satisfied easily in the azimuthally averaged state.

Substituting Eq. (5) into Eq. (8), the condition of gradient wind balance approximation can be written as Eq. (12),

This condition is more complex than that in the vertical direction. In order to better explain this condition of gra-dient wind balance approximation, three situations (at the low, middle, and top levels) are discussed as examples.

Considering that on the streamline of secondary circulation in the low-level eyewall, there is a point where the radial velocity value turns from negative to positive, and the tangent line of the streamline at this point is vertical, i.e., $ \beta =0 $. Assuming $ \left|{V}_{rz}\right|=\lambda v $ ($ \mathrm{\lambda } $ denotes a scale ratio), Eq. (12) can be written as Eq. (13),

On the TC scales (Willoughby, 1979), the ratio of $ \left|{\boldsymbol{V}}_{rz}\right| $ to v is about 0.2 ($ \lambda \approx 0.2 $). Therefore, the condition of gradient wind balance approximation is $ \left|\kappa \right| < 3.3\times {10}^{-4}\; {\mathrm{m}}^{-1} $ or $ \left|R\right| > $ 3 km, where $ R=\dfrac{1}{\kappa } $ indicates the curvature radius. However, there is usually a large curvature when the inflow in the low-level eyewall turns into updrafts sharply. For example, in terms of the TC simulated by Smith et al. (2009), u from 0 to w (equivalent to β from 0° to 45°) was achieved within half a kilometer (see Fig. 5e in their paper). Assuming that the curvature is uniform during this variation, we can obtain $ \kappa > 7.1 \times $$ {10}^{-4}\; {\mathrm{m}}^{-1} $ (R < 1.4 km). Thus, the gradient wind balance approximation is usually not satisfied in the low-level eyewalls of TCs.

For the updrafts of TC eyewall in the middle troposphere, the vertical acceleration is small and can be neglected, and then Eq. (13) can be written as Eq. (14),

Equation (14) denotes that the critical curvature depends not only on $ r $ and $ \lambda $, but also on $\; \beta $. For $\; \beta $ of 30°, 45°, and 60°, the conditions satisfying the gradient wind balance approximation are $ \left|\kappa \right| < 2.9\times {10}^{-4}\;{\mathrm{m}}^{-1} $, $ \left|\kappa \right| < 2.4\times$$ {10}^{-4}\;{\mathrm{m}}^{-1} $, and $ \left|\kappa \right| < 1.7\times {10}^{-4}\;{\mathrm{m}}^{-1} $, respectively. These conditions are stricter than those at the low level. The variable $ \beta $ is related to the slope of the streamline. Since the absolute angular momentum (M) is conserved along the streamline, the slope of the streamline is equivalent to that of the M surface. The slope equation of the M surface was proposed by Emanuel (1986), and its simplified form was derived by Stern and Nolan (2009). This slope equation indicates that the slope of the streamline increases linearly with the radius at constant pressure. This result suggests that $ \beta $ is larger for the streamline with a larger radius at the middle level, and the condition of gradient wind balance approximation is stricter. How-ever, according to the structure of TC secondary circulation shown by Liu et al. (1999), the streamline of middle-level updraft is far flatter than that of the low-level updraft. Therefore, the gradient wind balance approximation may still be satisfied at the middle level.

For the updrafts at the top of the TCs, $ \beta $ is close to 90°, implying that $ \mathrm{cos}\beta \approx 0 $ and $ \mathrm{sin}\beta \approx 1 $. Thus, at the top level, Eq. (12) is more complex to discuss since the denominator is close to 0. Multiplying Eq. (12) by $ \mathrm{cos}\beta /{\left|{\boldsymbol{V}}_{rz}\right|}^{2} $, we can obtain Eq. (15),

Due to the outward slope of updrafts and the conservation of M, $ r $ increases in the top of the TCs, and $ v $ decreases. Adding $ \mathrm{cos}\beta $, the right side of Eq. (15) is close to 0. Under the assumption of axisymmetry and steady state, the vertical acceleration on the left side is replaced by Eq. (10). Hence, the condition of gradient wind balance approximation can be expressed as Eq. (16),

The condition of gradient wind balance approximation can be fixed for a specified vertical acceleration (not depending on $ \kappa $). Assuming that $ w $ decreases steadily from $ \left|{\boldsymbol{V}}_{rz}\right|$cos45° to $ 0 $ within 8 km (from the middle level to the top level), the condition of gradient wind balance approximation at the top level is $ \kappa \approx 3.1\times {10}^{-5}\;{\mathrm{m}}^{-1} $. This condition requires coordination between the radial and vertical momentums, considerably stringent than that at the middle and low levels.

Using the relationship equation between curvature and acceleration, we demonstrate that the radial and vertical acceleration on the TC secondary circulation streamline cannot be zero simultaneously, i.e., the thermal wind balance cannot be strictly satisfied on the curve streamline. Based on this relationship, we also discuss the conditions of thermal wind balance approximation in TC eyewall. The results show that the hydrostatic balance approximation is satisfied easily at the azimuthally averaged state. However, there is a limitation on the curvature of the streamline, which allows the gradient wind balance approximation in the eyewall. The limitation becomes stricter as altitudes increase. The TC scales provided by Willoughby (1979) are used to discuss the conditions of gradient wind balance approximation at the low, middle, and top levels of the TC eyewall. The results indicate that the gradient wind balance approximation is invalid in the low-level eyewall since there is usually a large curvature when the inflow in the low-level eyewall turns into updrafts sharply. In addition, it also is invalid at the top level because the highly strict limitation of the streamline curvature is too easily broken.

Smith et al. (2009) emphasized the gradient wind imbalance in the low-level eyewall, and Cohen et al. (2017) demonstrated that it also exists at the top level through simulation. The gradient wind imbalance at the top level of TCs implies that the highly strict conditions are easily broken. Bryan and Rotunno (2009) showed two imbalance patterns of the secondary circulation of a simulated TC, which corresponded well to the curvature of the streamline. This finding is in good agreement with the conclusions of this research.

In this study, we propose an integral pattern for the thermal wind balance of TC secondary circulation in a physical perspective. The validation of the balance approximation is related to the curvature of the secondary circulation streamline, suggesting that the strict thermal wind balance is incompatible with the curve secondary circulation streamline. However, this incompatibility is ignored in some TC structure theories. For example, the TC theory proposed by Emanuel (1986) is based on the thermal wind balance, but it predicted curve secondary circulation. Therefore, it is necessary to further investigate the TC structure theory.

The authors would like to thank Dr. Tim Dunkerton of the NorthWest Research Associates for his helpful suggestions on this manuscript.