Tropical cyclone size is strongly limited by the Rhines scale: experiments with a barotropic model
NOT PUBLISHED. Submitted for peer review

The size of a tropical cyclone (TC) determines its footprint of gale-force winds (Powell and Reinhold 2007), storm surge (Irish et al. 2011) and rainfall (Kidder et al. 2005; Lavender and McBride 2021). Therefore, understanding the dynamics of TC size is important for understanding potential TC impacts.

Observational studies have found that TC size can vary significantly. For example, the TC radius of vanishing wind ($R_{0}$) typically ranges from $400$km to $1100$km (Chavas et al. 2016). Past studies have shown that TC size may be sensitive to a variety of parameters, such as synoptic interaction (Merrill 1984; Chan and Chan 2013), time of the day (Dunion et al. 2014), environmental humidity (Hill and Lackmann 2009), and latitude (Weatherford and Gray 1988a, b; Chavas et al. 2016). On the f-plane, TC size decreases with decreasing Coriolis parameter $f$ following an $f^{-1}$ scaling (Chavas and Emanuel 2014; Khairoutdinov and Emanuel 2013; Zhou et al. 2014), suggesting that size should decrease rapidly with latitude in the tropics. However, in observations, TCs size tends to increase slowly with latitude (Kossin et al. 2007; Knaff et al. 2014). Indeed, Chavas et al. (2016) showed explicitly that the inverse-f dependence can not explain the observed dependence of TC size with latitude.

Recently, Chavas and Reed (2019, hereafter CR19) used aquaplanet experiments with uniform thermal forcing to demonstrate that median TC size scales with the Rhines scale (Rhines 1975). This scale depends inversely on the planetary vorticity gradient, $\beta$, and increases very slowly with latitude in the tropics, which matches the behavior seen in observations. Their findings lead to the question: is the size of an individual storm limited by the Rhines scale and, if so, why?

The Rhines scale has traditionally served as the scale that divides flow into turbulence and Rossby wave-dominated features (Held and Larichev 1996). When the eddy length scale is at or larger than the Rhines scale, and the linear Rossby wave term dominates the nonlinear turbulent term. Conceptually, this implies that an eddy circulation larger than the Rhines scale behaves more wave-like. Despite being used to understand the role and scale of eddies in many large-scale atmospheric and oceanic circulation theories (e.g., James and Gray 1986; Vallis and Maltrud 1993; Held and Larichev 1996; Held 1999; Lapeyre and Held 2003; Schneider 2004; LaCasce and Pedlosky 2004), it has not been applied to understand the scale of the tropical cyclone. Since a TC can be regarded as an eddy circulation embedded in the tropical atmosphere, it seems plausible that the Rhines scale can indeed directly modulate TC size. However, it is not clear how $\beta$ acts to limit the size of an individual TC and how this may be understood in the context of the Rhines scale.

The Rhines scale is typically calculated using a single characteristic turbulent velocity, usually defined as the root-mean-square velocity (e.g., Sukoriansky et al. 2006; Kidston et al. 2010), or even a single characteristic velocity scale for a TC (an outer circulation velocity scale $U_{\beta}$ in CR19; a collective velocity scale in Hsieh et al. 2020). However, a TC clearly does not possess a single velocity scale but instead has rotational velocities that vary strongly as a function of radius. Moreover, a theoretical model now exists for the radial structure of the TC wind field that captures the first-order behavior of TC structure found in observations (Chavas et al. 2015; Chavas and Lin 2016). Such a wind field model may be used to understand the detailed dynamics of this Rhines scale effect within a TC.

The simplest way to test the effect of $\beta$ on a TC-like vortex is to perform barotropic model simulations of a single TC-like vortex on a $\beta$-plane, as the low-level circulation of a TC may be considered approximately barotropic and the Rhines scale arises from the barotropic vorticity equation itself. This approach neglects the role of the secondary circulation, which is undoubtedly an important part of TC dynamics, in order to isolate the basic behavior and dynamical response of the primary circulation of a TC. Various aspects of the dynamics of a vortex on a $\beta$-plane have been analyzed in past studies. The most widely-known effect is $\beta$-drift (Chan and Williams 1987), which is the poleward and westward vortex translation induced by the interaction of the vortex and the vortex-generated planetary Rossby waves¹¹1We use the term “planetary” to emphasize that these Rossby waves arise due to vortex flow across the meridional vorticity gradient of the Earth’s rotation, $\beta$. This is in contrast to “vortex Rossby waves” [Montgomery and Kallenbach 1997], which are Rossby waves that arise from vortex flow across the radial relative vorticity gradient of the vortex itself. (Llewellyn Smith 1997; Sutyrin and Flierl 1994; Fiorino and Elsberry 1989; Wang et al. 1997). Notably, though not emphasized in their study, Chan and Williams (1987) demonstrated in their $\beta$-drift experiments that vortex size tends to decrease with time on a $\beta$-plane. While inducing translation, these Rossby waves transfer kinetic energy from vortex to Rossby waves that then propagate into the environment (Flierl and Haines 1994; Sutyrin et al. 1994; Smith et al. 1995), thereby weakening the primary circulation and hence reducing vortex size (Mcdonald 1998; Sai-Lap Lam and Dritschel 2001; Eames and Flór 2002). Eames and Flór (2002) found that for larger vortex size, the Rossby wave generation will dominate the dynamics of the vortex and, further, the vortex translation speed is correlated with the Rossby waves phase speed, a result that is conceptually similar to the Rhines scale effect described above. However, past work has yet to systematically test and explain the response of the size of an individual TC to $\beta$ and place it in the context of the Rhines scale.

Here we focus on understanding the detailed response of the structure and size of an individual TC-like vortex to $\beta$. Our principal research questions are:

1. 1.

   How does the size of and individual TC-like vortex respond to $\beta$?

2. 2.

   Can we develop a framework explaining why $\beta$ limits storm size and its relationship to the traditional Rhines scale?

3. 3.

   Can we predict the time-dependent response of TC size to $\beta$?

To answer these questions, we first revisit the meaning of the Rhines scale in the context of an individual coherent vortex and show how it may be re-expressed in the context of a vortex with a known wind profile that is useful for understanding the response of an axisymmetric vortex to $\beta$. We then use this theory to analyze dynamical experiments using a simple barotropic model on a $\beta$-plane initialized with the axisymmetric low-level tropical cyclone wind field defined in Chavas et al. (2015). We conduct experiments systematically varying $\beta$ and initial vortex size to investigate the detailed time-dependent response of the vortex to the $\beta$. Overall, our focus is on understanding the nature of the response of the size of a TC-like vortex to $\beta$ and how we may use the conceptual foundation of the Rhines scale to predict it. Analysis of the details of energy transfer between the vortex and Rossby waves is left for future work.

The paper is organized as follows: Section 2 presents the theory and proposes our hypotheses; Section 3 demonstrates our model configuration and experiment designs; Section 4 presents the idealized results and analyses of our experiments; Section 5 presents our key findings and discusses the implications of our results and their relationship to real TCs on Earth.

The goal of this study is to investigate the limitation of vortex size by the Rhines scale, which is a parameter that governs the interaction between Rossby waves and a vortex in a fluid. This effect exists in any fluid in the presence of a planetary vorticity gradient, $\beta=\frac{\partial f}{\partial y}$, where $f$ is the Coriolis parameter and $y$ is the meridional direction. The simplest such system is a dry barotropic (i.e. single-layer) fluid with constant depth. Such a fluid will obey the non-divergent dry barotropic vorticity equation as following:

where $\partial$/$\partial t$ is the Eulerian tendency, $\zeta$ is the relative vorticity, $\overrightarrow{\textbf{u}}$ is horizontal wind velocity, $v$ is the meridional wind speed, and $\beta$ is the meridional gradient of planetary vorticity. The term on the left hand side is the vorticity tendency, the first RHS term is the non-linear advection of relative vorticity (hereafter ”non-linear term”), and the second RHS term is the linear advection of the planetary vorticity (hereafter ”$\beta$ term”). Note that, unlike a shallow-water system, this system has no gravity waves because the fluid depth is constant.

To determine which term in the barotropic vorticity equation dominates the vorticity tendency, a traditional scale analysis of Eq. 1 would yield:

where the $V$ is the speed scale of the wind, $L$ is the horizontal length scale, $T$ is the time scale. For a non-divergent axisymmetric vortex, $\overrightarrow{\textbf{u}}$ and $v$ can be both expressed as the azimuthal wind speed of the vortex circulation $U_{c}$. Note that the advection term in Eq. 2 has $L^{2}$ in denominator, but each $L$ has a different physical meaning: the relative vorticity, $\zeta=\partial(rU_{c})/dr$, represents the radial gradient of $U_{c}$, and hence $L_{\zeta}=R$; in contrast, the advection operator, $\overrightarrow{\textbf{u}}\cdot\bigtriangledown$, is the tangential advection around the circumference of the vortex, and hence $L_{\overrightarrow{\textbf{u}}\cdot\bigtriangledown}=2\pi R$. Together, in Eq. 2, the denominator becomes $L^{2}=2\pi R^{2}$. Therefore, the ratio between the non-linear advection term and the $\beta$ term, which we define as the Rhines number ($Rh$), can be written as following:

Eq. 3 neglects any radial flow as is required for a barotropic vortex. Real TCs possess significant inflow at low-levels, which is a topic we address in the discussion of our results below.

When $Rh=1$, the non-linear term and the $\beta$-term are equal (Vallis 2017, p.446). We can rearrange Eq. 3 to define the Rhines scale ($R_{Rh}$) for a rotational flow with speed of $U_{c}$:

Note that the key distinction of Rhines scale from a deformation-type scale is that the velocity scale is a true flow speed rather than a gravity wave phase speed. We emphasize that we are precise in including the $2\pi$ factor in our explanation above, as it is quantitatively important for our results presented below. This is in contrast to typical scale analyses which are agnostic to the inclusion or neglect of constant factors (and indeed the appearance of such factors varies in the literature).

Previous studies tend to estimate a system’s Rhines scale by assuming a characteristic eddy flow velocity (e.g., CR19; Chemke and Kaspi 2015; Frierson et al. 2006), such that the system will have a single characteristic Rhines scale. However, an individual TC-like vortex possesses circulation speeds that vary strongly with radius and hence contains both ”small” and ”large” circulations simultaneously by definition. Thus, the details of these wave effects should depend on radius and cannot be characterized by a single velocity nor a single Rhines length scale. The simplest starting point is to consider the vortex as comprised of independent circulations at each radius with different wind speeds and calculate their corresponding Rhines scale as a function of radius. This results in a radial profile of $R_{Rh}$ that depends on the vortex’s tangential wind profile and $\beta$ (eq. 4). Here we use the wind speed of the TC’s circulation ($U_{c}$), which varies with radius, to calculate the radial profile of the Rhines scale in a TC. This allows one to evaluate how $\beta$ may affects the vortex circulation differently at different radii.

For a sufficiently large circulation relative to the Rhines scale ($R\gg R_{Rh}$), the non-linear term is small ($Rh\ll 1$) and thus its vorticity tendency will be governed by the $\beta$ term, which gives pure Rossby waves. Therefore, a circulation at a given radius that is larger than its corresponding Rhines scale will be affected by $\beta$ and generate Rossby waves, which will radiate energy away. Meanwhile, for a sufficiently small circulation relative to the Rhines scale ($R\ll R_{Rh}$), the $\beta$ term is small ($Rh\gg 1$), and thus its vorticity tendency will be governed by the non-linear term and will generate minimal Rossby wave activity. For a perfectly axisymmetric vortex, the relative vorticity advection term is zero, and hence the circulation simply circulates without the energy sink from Rossby waves.

The concept of the Rhines scale for a vortex can be more easily understood if rephrased in terms of a velocity scale rather than length scale. Since $R_{Rh}$ is function of $U_{c}$ (eq.4), that means for a given wind speed, $R_{Rh}$ defines a specific length scale whose value relative to the circulation radius determines whether the circulation will generate significant Rossby waves or not. Alternatively, one may choose to fix the circulation radius, in which case $Rh$ equivalently defines a specific wind speed whose value relative to the wind speed at radius determines whether the circulation will be affected by Rossby waves or not. Based on this concept, we can rearrange the Eq. 3 to yield the wind speed when $Rh=1$, which we define as the Rhines speed ($U_{Rh}$):

Importantly, the radial dependence of $U_{Rh}$ is solely a function of $\beta$, and for a given value of $\beta$, its profile is fixed and is independent of the vortex. The case $U_{c}\ll U_{Rh}$ is analogous to $R\gg R_{Rh}$ and corresponds to the wave-dominant regime, while the case $U_{c}\gg U_{Rh}$ is analogous to $R\ll R_{Rh}$ and corresponds to the vortex-dominant regime.

The above definition is especially convenient because the radial structure of the circulation may be known or specified. Figure 1 displays an example radial profile of $U_{c}$ (blue solid line) for a tropical cyclone, defined by the model for the low-level azimuthal wind of Chavas et al. (2015; described in detail in Section 3 below), and the $U_{Rh}$ (blue dashed line) and $R_{Rh}$ (brown dashed line) that are calculated using Eqs. 5 and 4, respectively. Typically, a TC-like $U_{c}$ profile will decrease monotonically with radius outside the radius of maximum wind. Meanwhile, the $U_{Rh}$ (blue dashed line) profile increases with radius monotonically. Therefore, there will be an intersection between these two curves at a specific radius, which we define as the vortex Rhines scale (${R}_{VRS}$, red vertical line). For the convenience of later discussion and analyses, we also define the value of $U_{c}$ at $R_{VRS}$ as the vortex Rhines speed ($U_{VRS}$, green horizontal line). We may further define the turn-over time scale of the circulation at $R_{VRS}$ as the vortex Rhines timescale ($T_{VRS}$), which can be written as:

Analogous to the traditional Rhines scale separating wave and vortex dominant regimes, ${R}_{VRS}$ divides the vortex into a vortex-dominant region at smaller radii and a wave-dominant region at larger radii, as shown by Figure 1, which we define as the ”wave region” and ”vortex region”. In the vortex region at smaller radii, planetary Rossby waves are not readily generated and the rapid rotation will axisymmetrize the vorticity field (Montgomery and Kallenbach 1997) until the vortex flow is parallel to the vorticity contours and the flow become quasi-steady (Eq. 1). Meanwhile, in the wave region at larger radii, the rotating flow is slow enough to generate significant planetary Rossby wave activity, which will cause asymmetric deformation of the vortex flow at those radii. Taken together, the expectation is that only the circulation in wave-dominant region will stimulate significant Rossby waves, distorting and gradually dissipating the circulation therein. Meanwhile, the circulation in the vortex-dominant region would be expected to remain nearly steady. As a result, vortex size will be limited by ${R}_{VRS}$.

To investigate how ${R}_{VRS}$ affects the size of a TC-like vortex, we propose following hypotheses:

1. 1.

   Vortex size is limited by its vortex Rhines scale, ${R}_{VRS}$.

2. 2.

   A larger/smaller vortex relative to its vortex Rhines scale will shrink faster/slower.

Below we test these hypotheses by simulating a TC-like vortex on a $\beta$-plane. We focus here on characterizing and understanding the vortex response to $\beta$ across experiments varying TC size and $\beta$. Theoretical analysis of the detailed energetics of this response is left to future work.

This study derives a new concept called the vortex Rhines scale and applies it to experiments with a barotropic model to understand how $\beta$ limits the size of a tropical cyclone-like vortex. Since the barotropic model governing equation includes only the advection of the relative and planetary vorticity, our experiment design provides an idealized and straightforward framework to investigate the dynamics of a TC-like vortex in the presence of $\beta$ and its relationship to the traditional Rhines scale.

The key findings of this study are as follows:

1. 1.

   We derive a quantity called the vortex Rhines scale ($R_{VRS}$), which translates the traditional Rhines scale into the context of an individual axisymmetric vortex, and show how it can be used to understand the effect of $\beta$ on a TC-like vortex.

2. 2.

   The vortex Rhines scale serves as a robust limit on the size of a TC-like vortex on a barotropic $\beta$-plane, which corroborates the finding in Chavas and Reed (2019) that storm size scales with the traditional Rhines scale. The circulation beyond the vortex Rhines scale will weaken with time, which manifests itself as a shrinking of vortex size. $R_{VRS}$ offers a more useful scale for the limit of TC size than the traditional Rhines scale.

3. 3.

   Theoretically, the vortex will be divided into two regions by $R_{VRS}$: vortex region at smaller radii and wave region at larger radii. In the vortex region, the circulation is highly axisymmetric and largely unaffected by $\beta$. In the wave region, planetary Rossby wave generation is strong and waves distort and dissipate the outer circulation.

4. 4.

   A larger vortex relative to its $R_{VRS}$ will shrink faster, and all vortices shrink towards an equilibrium close to its vortex Rhines scale.

5. 5.

   Vortex size shrinks toward $R_{VRS}$ following a dominant timescale given by $T_{VRS}$, though the role of that timescale differs slightly when varying $R_{0}$ vs $\beta$. $T_{VRS}$ is also shown to be closely related to the Rossby wave group velocity at the vortex Rhines scale, which provides a direct link between our theory and the dynamics of the waves themselves.

6. 6.

   The first-order evolution of the vortex for any value of $R_{0}$ and $\beta$ is controlled by the initial value of $R_{VRS}$ and $T_{VRS}$, thereby enabling one to predict the vortex response from the initial condition alone.

7. 7.

   A similar outcome occurs when allowing $f$ in the initial vortex structure to change consistent with $\beta$, indicating that the results are also applicable to an Earth-like setting.

It is important to place our idealized results in the context of real TCs on Earth. The typical duration of a TC is on the order of 10 days (Webster et al. 2005) and form from pre-existing disturbances that may have propagated for much longer. Since most of our experiment members (especially for those on a lower latitude $\beta$-plane) have shrunk rapidly within 10 days, the vortex Rhines scale would be expected to strongly limit TC size on Earth and hence explain why storm size appears to follow a Rhines-type scaling in observations and models. Moreover, it explains how storm size may vary widely in nature, as TC size may remain steady at any size that is reasonably small relative to this scale, since the vortex Rhines scale only sets an upper bound on vortex size. Additionally, $R_{VRS}$ in our experiments range from $200$ to $400$ km, which is substantially smaller than the $f^{-1}$ theoretical length scale for TC size on the $f$-plane, $V_{p}/f$, where $V_{p}$ is the potential intensity (Chavas and Emanuel 2014), which is larger than $1000$ km at low latitudes (and goes to infinity at the equator). In other words, if a TC were for form with a size equal to $V_{p}/f$, our results indicate that $\beta$ and its induced wave effects would cause it to shrink rapidly within a few days. This is a simple mechanistic explanation for why $V_{p}/f$ is not an appropriate scaling for TC size in the tropics as has been noted in observations (Chavas et al. 2016). Finally, though our derivations and analyses have assumed axisymmetry, these physics may be general to any vortex, even asymmetric ones such as extra-tropical cyclones. Indeed, our results and theory suggests a simple mechanistic explanation for why the Rhines scale cuts of the upscale energy cascade in 2D turbulence and limits extra-tropical cyclone size (e.g., Held and Larichev 1996; Chai and Vallis 2014; Chemke and Kaspi 2015, 2016; Chemke et al. 2016).

All of our experiments are barotropic. This simplified approach omits various key physical features in real TCs, but it is unclear how non-barotropic effects would modify these barotropic responses. For example, radial inflow in TC will communicate the reduction in angular momentum in the outer region to the inner core, so the inner core wind field would be expected to shrink too. Further investigation in models with higher degrees of complexity and in observations is needed to evaluate our findings in a full-physics baroclinic environment.

Finally, to provide a broader perspective on our findings, we conclude with a simple conceptual diagram to place the effect of $\beta$ in the context of standard dynamical balance regimes common in atmospheric science. Figure 10 illustrates different dynamical regimes as a function of radius for an example TC wind profile. On an $f$-plane ($\beta=0$), a vortex wind profile can remain steady at all radii and its size remain unchanged. This is because on the $f$-plane at any given radius the flow can exist in a balanced state, where the specific balance is commonly defined using the Rossby number. Within the inner core region (inside of approximately 100 km in Figure 10), the Rossby number of the circulation is significantly larger than 10, which corresponds to cyclostrophic balance. Between the inner core and far outer circulation, the Rossby number ranges between 0.1 and 10, which corresponds to gradient wind balance. Finally, in the far outer circulation inside of the outer radius, the Rossby number is less than 0.1, which corresponds to geostrophic balance. However, in the presence of non-zero $\beta$, there is now an unbalanced regime at radii beyond the vortex Rhines scale, which corresponds to the wave region described above. In this unbalanced region, the outer circulation stimulates planetary Rossby waves, distorting the flow and transferring kinetic energy out of the vortex.