Dynamical constraints on the ITCZ extent in planetary atmospheres

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Dynamical constraints on the ITCZ extent in planetary atmospheres
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Sean Faulk⇤
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Earth, Planetary, and Space Sciences, UCLA, Los Angeles, California, USA
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Jonathan Mitchell
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Earth, Planetary, and Space Sciences, Atmospheric and Oceanic Sciences, UCLA, Los Angeles,
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California, USA
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Simona Bordoni
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Caltech, Pasadena, CA, USA
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⇤ Corresponding
author address: Department of Earth, Planetary, and Space Sciences, University
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of California Los Angeles, 595 Charles Young Drive East, Los Angeles, California 90095, USA.
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E-mail: spfaulk@ucla.edu
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ABSTRACT
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We study a wide range of atmospheric circulations with an idealized moist
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general circulation model to evaluate the mechanisms controlling intertrop-
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ical convergence zone (ITCZ) migrations. We employ a zonally symmetric
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aquaplanet slab ocean of fixed depth and force top-of-atmosphere insolation
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to vary seasonally as well as remain fixed at the pole in “eternal solstice” runs.
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We explore a range of surface heat capacities and rotation rates, keeping all
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other parameters Earth-like. For rotation rates WE /8 and slower, the seasonal
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ITCZ reaches the summer pole. Additionally, in contrast to previous ther-
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modynamic arguments, we find that the ITCZ does not follow the maximum
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moist static energy, remaining at low latitudes in the “eternal solstice” case for
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Earth’s rotation rate. Furthermore, we find that significantly decreasing heat
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capacity does little to extend the ITCZ’s summer migration off the equator.
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These results suggest that the ITCZ may be more controlled by dynamical
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mechanisms than previously thought; however, we also find that baroclinic
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instability, often invoked as a limiter on the extent of the annual and summer
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Hadley cell, appears to play little to no role in limiting the ITCZ’s migra-
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tion. We develop an understanding of the ITCZ’s position based on top-of-
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atmosphere energetics and boundary layer dynamics and argue that friction
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and pressure gradient forces determine the region of maximum convergence,
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offering a new perspective on the seasonal weather patterns of terrestrial plan-
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ets.
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1. Introduction
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One of the most prominent features of Earth’s large-scale atmospheric circulation in low lati-
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tudes is the intertropical convergence zone, or the ITCZ (Waliser and Gautier 1993), defined in
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this study as the latitude of maximum zonal mean precipitation. Associated with the ascending
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branch of the Hadley cell, the ITCZ is the region of moist uplift, and thus of high precipitation and
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deep convection, that occurs where the low-level winds of the two cells converge. During summer,
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this convergence zone migrates off the equator into the summer hemisphere, up to 10N in oceanic
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regions and occasionally as far as 30N in the case of the Asian summer monsoon (Yihui and Chan
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2005).
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The mechanisms that control the ITCZ, though, remain unclear. Ideally, the convergence occurs
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over the warmest surface waters. Indeed, in the northeast Pacific ocean, the ITCZ remains north
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of the equator throughout the entire year, coinciding largely with high sea surface temperatures
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(SSTs), which remain north of the equator even during northern winter (Janowiak et al. 1995).
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However, previous observations show that the ITCZ does not always coincide with the SST max-
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imum and corresponding local sea level pressure minimum (Ramage 1974; Sadler 1975). In light
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of such observations, and given the importance of moisture transport to tropical communities,
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much attention has been paid in the literature towards identifying and understanding controls on
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the ITCZ (Waliser and Somerville 1994; Sobel and Neelin 2006; Schneider et al. 2014). Much
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of this work can be roughly divided into two theories for understanding the ITCZ: one based on
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thermodynamics and the other on momentum dynamics.
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a. Thermodynamic theory
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Thermodynamic theories posit that deep convection is thermodynamically controlled. Conver-
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gence and precipitation are determined by local vertical temperature and moisture profiles gov3
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erned by the moisture budget and surface and radiative fluxes (Sobel and Neelin 2006). When
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applied with the “weak temperature gradient” approximation, some such theories successfully re-
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construct tropical convergence and tend to predict maximum convergence and rainfall over the
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warmest surface temperatures (Neelin and Held 1987; Sobel and Bretherton 2000). When applied
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to convergence zones associated with zonally averaged overturning circulations and coupled to
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quasi-equilibrium theories of moist convection and its interaction with larger-scale circulations,
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these thermodynamic constraints argue that the ITCZ lies just equatorward of the maximum low-
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level moist static energy (MSE) (Emanuel et al. 1994; Lindzen and Hou 1988; Privé and Plumb
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2007; Bordoni and Schneider 2008). For this reason, the distribution of MSE has been extensively
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used as a diagnostic of the ITCZ position.
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However, the ITCZ has also been shown to respond strongly to extratropical thermal forcings,
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with shifts of the ITCZ into (away from) a relatively warmed (cooled) hemisphere, indicating that
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forcings remote from precipitation maxima can influence the tropical circulation and convergence
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zones (Chiang and Bitz 2005; Broccoli et al. 2006). Such work reveals the role of the vertically
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integrated atmospheric energy budget in ITCZ shifts and has led to the development of another
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diagnostic, the energy flux equator, which is the latitude at which the zonal mean MSE meridional
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flux vanishes (Kang et al. 2008). In recent studies focused on global energy transports, the energy
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flux equator has been shown to be well correlated with the ITCZ’s poleward excursion into the
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summer hemisphere, and the energetic framework in general has proved useful for understanding
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the ITCZ (Kang et al. 2008, 2009; Chiang and Friedman 2012; Frierson and Hwang 2012; Frierson
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et al. 2013; Bischoff and Schneider 2014). Lastly, through its control on thermodynamic quantities,
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heat capacity has also been shown to impact the ITCZ, with convergence generally traveling farther
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poleward over surfaces of lower heat capacity (Fein and Stephens 1987; Xie 2004; Donohoe et al.
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2014; Bordoni and Schneider 2008).
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b. Dynamic theory
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Dynamic theories use the boundary layer momentum budget to determine winds and hence con-
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vergence, showing that convection is driven primarily by boundary layer (BL) momentum dynam-
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ics. Perhaps the most influential of these studies, Lindzen and Nigam (1987) showed that the
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pressure gradients associated with even the small surface temperature gradients of the tropics have
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a substantial impact on low-level flow and convergence. Their model, built on tropical SST gra-
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dients, adequately reproduces the observed tropical convergence, though the model’s assumptions
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have been debated. Back and Bretherton (2009) address such concerns by testing and expanding
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upon Lindzen and Nigam (1987), showing that SST gradients in reanalysis data are indeed consis-
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tent with BL convergence, which in turn causes deep convection. Along a similar vein, Tomas and
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Webster (1997) found that in regions where the surface cross-equatorial sea level pressure gradient
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was weak, convection tended to coincide with the maximum SST, but in regions with a substan-
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tial cross-equatorial pressure gradient convection tended to lie equatorward of the maximum SST,
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also suggesting the importance of temperature gradients, as opposed to maxima, in driving the
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flow. Subsequent studies develop a theory to determine the location of the ITCZ based on the
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cross-equatorial pressure gradient, which drives anticyclonic vorticity advection across the equa-
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tor to render the system inertially unstable (Tomas and Webster 1997; Tomas et al. 1999; Toma
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and Webster 2010). Thus, convergence and divergence are shown to be separated at the latitude
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where the zonal mean absolute vorticity is equal to zero.
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c. ITCZ on terrestrial planets
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Examining the behavior of the ITCZ on other terrestrial planets could provide further insight into
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the mechanisms that control its movement. It has been argued, using general circulation models
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(GCMs), that analogous regions of seasonal convergence and ascent associated with the Hadley
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cell exist on other planetary bodies, namely Mars (Haberle et al. 1993; Lewis 2003) and Titan
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(Mitchell et al. 2006). The ascent region migrates significantly off the equator into the summer
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hemisphere during summer solstice, perhaps even reaching the summer pole on Titan. Indeed,
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there have been multiple observations of methane cloud activity near Titan’s south pole during
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southern summer solstice (Bouchez and Brown 2005).
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Factors that control the movement of the ITCZ—particularly what prevents it from extending
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too far beyond the equator on Earth yet allows it to reach the summer pole on Titan—still remain
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unclear. The far-reaching solstitial Hadley cells on Mars and Titan are often ascribed to the planets’
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low surface heat capacities due to their lack of global oceans. But the extent to which the smaller
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rotation rate of Titan and the smaller radii of both planets may also contribute to the seasonality
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of their convergence zones is unknown. From classical dry axisymmetric theory, the width of the
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Hadley cell expands with decreasing rotation rate and/or decreasing radius (Held and Hou 1980;
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Caballero et al. 2008). Many parameter space studies using GCMs (Williams 1988; Navarra and
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Boccaletti 2002; Walker and Schneider 2006; Mitchell and Vallis 2010; Mitchell et al. 2014; Pinto
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and Mitchell 2014; Kaspi and Showman 2015) show that slowly rotating planets exhibit expanded
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Hadley cells and effectively become “all-tropics” planets similar in circulation structure to Titan
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(Mitchell et al. 2006). But no parameter space studies have been done using a moist GCM with a
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seasonal cycle, and focusing on the ITCZ, as we do in this study.
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In addition, the classical axisymmetric theory does not account for the effect of baroclinic eddies
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(Schneider 2006). GCM studies have highlighted the influence of eddy momentum fluxes on the
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Hadley circulation by studying the response of Earth’s Hadley circulation to seasonal transitions
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(Walker and Schneider 2005; Bordoni and Schneider 2008; Schneider and Bordoni 2008; Merlis
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et al. 2013) and the extent of the annual-mean Hadley circulation over a range of planetary pa-
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rameters (Walker and Schneider 2006; Levine and Schneider 2011). It has been suggested that
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the annual-mean Hadley circulation width may be affected by extratropical baroclinic eddies that
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limit its poleward extent (Schneider 2006; Levine and Schneider 2015). Though no work has ex-
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tended this argument to consideration of how baroclinic eddies might influence the circulation, the
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above studies show that eddy momentum fluxes impact the tropical overturning in equinoctial and
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solstitial circulations; thus, the possibility that baroclinic eddies, by strongly impacting the width
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and strength of the summer cell, may influence the ITCZ deserves investigation.
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We examine these many potential controls on the ITCZ’s movement using a moist Earth GCM
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of varying heat capacities, seasonal forcings, and rotation rates. The primary controls in question
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are that of heat capacity, low-level MSE, baroclinic instability, atmospheric energy balance, and
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BL dynamics. In section 2, we describe the model and experimental setup. In section 3 and 4, we
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evaluate theories relying on heat capacity, maximum MSE, and baroclinic instability. In section 5,
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we analyze the general circulation of our rotation rate experiments. Then in section 6, we present
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analysis of the BL dynamics and radiative energy balance for all experiments.
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2. Methods
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We use the moist idealized three-dimensional GCM described in Frierson et al. (2006) and Frier-
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son (2007) based on the Geophysical Fluid Dynamics Laboratory (GFDL) spectral dynamical core,
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but with two major changes: 1) the addition of a seasonal cycle, and 2) long-wave optical depths
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that do not depend on latitude.
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Radiative heating and cooling are represented by gray radiative transfer, in which radiative fluxes
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are only a function of temperature, thus eliminating water vapor feedback. There is no diurnal
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cycle. The convection parameterization is a simplified Betts-Miller scheme, described fully in
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Frierson (2007). The scheme relaxes the temperature and moisture profiles of convectively un-
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stable columns to a moist adiabat with a specified relative humidity (70% in these simulations)
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over a fixed relaxation time (2 hours). Standard drag laws are used to calculate surface fluxes,
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with drag coefficients determined by a simplified Monin-Obukhov scheme. The BL scheme is a
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standard K-profile scheme with diffusivities consistent with the simplified Monin-Obukhov the-
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ory. The lower boundary is a zonally symmetric slab mixed layer ocean with a constant depth of
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10 m in the control case, corresponding to a heat capacity C of 1 ⇥107 J m
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inertia timescale t f v20 days, where t f =
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surface temperatures are prognostic and adjust to ensure the surface energy budget is closed in the
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time-mean.
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a. Seasonal cycle
C
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4s T
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K
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and a thermal
and T = 285 K (Mitchell et al. 2014). Thus, sea
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We implement the seasonally varying top-of-atmosphere (TOA) insolation prescription from
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(Hartmann 1994, pp. 347-349), wherein the declination angle is approximated by a Fourier series
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that provides an empirical fit to Earth’s current insolation. The left panel of Fig. 1 shows the
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resulting seasonal cycle of insolation. The prescription does account for Earth’s eccentricity.
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b. Optical depth
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The latitude-dependent optical depth of the original model was suited for an equinoctial frame-
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work and is thus inappropriate for our seasonal framework where bands of precipitation and hu-
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midity fluctuate in latitude over the course of a year. In our model, the prescribed optical depth is
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independent of latitude, becoming a function only of pressure with a linear component, describing
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the effect of well-mixed greenhouse gases, and a quartic component, capturing the effect of water
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vapor confined to the lower troposphere. Parameters of the pressure-dependent optical depth are
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same as in Frierson et al. (2006), except the surface value of optical depth t0 = 12 (t0e + t0p ), where
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t0e and t0p represent the equator and pole, respectively.
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c. Experimental setup
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The model uses the primitive equations with T42 spectral resolution and 25 unevenly spaced
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vertical levels, with greater resolution in the BL. We conduct three primary sets of simulations: 1)
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we reduce the depth of the mixed layer (values shown in Table 1, Earth slab ocean depth as from
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Donohoe et al. (2014)) with all other parameters kept Earth-like (default parameters as in Frierson
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et al. (2006)) and the insolation cycle kept fixed; 2) we reduce the frequency of the seasonal cycle
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down to the extreme case where the planet is in “eternal solstice” (shown in the right panel of
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Fig. 1) with all other parameters kept Earth-like, done to observe ITCZ behavior over the longer
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timescales of fixed insolation; and 3) we adjust the rotation rate of an Earth-like planet from
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4 times larger than Earth’s value down to 32 times smaller than Earth’s value, where parameters
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other than rotation rate (insolation, radius, gravity, etc.) are kept Earth-like and the insolation cycle
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is kept fixed. In addition to these eddy-permitting simulations, we run axisymmetric simulations
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for end-member rotation rates, with both seasonal and “eternal solstice” forcings.
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All simulations are run for ten years. Values for the seasonal cases are composited over the last
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eight years (two years of spin-up). All solstitial time averages for each year of the seasonal cases
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are taken during a 20-day period centered around the time of maximum zonally-averaged northern
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hemisphere precipitation. Values for the “eternal solstice” simulations are averaged over the final
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year (nine years of spin-up).
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3. Evaluating thermodynamic mechanisms
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In this section, we address to what extent theories for the ITCZ position developed within the
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thermodynamic framework explain results in our simulated experiments. In particular, we focus
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on the impact of heat capacity and MSE on the ITCZ position.
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a. Heat capacity
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Ideally, the ITCZ moves farther over land than over ocean because of land’s lower heat capacity.
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The most notable manifestation of this is the Asian monsoon, Earth’s largest and most extensive
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monsoon that reaches up to 30N as it travels over the Indian subcontinent and mainland China
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(Yihui and Chan 2005). Furthermore, explanations for the Hadley cell’s larger migrations on Mars
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and Titan as compared to Earth often emphasize the lower surface heat capacities of those planets,
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since they lack global oceans. In an aquaplanet or slab ocean experiment, the heat capacity is
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synonymous with the ocean depth and previous slab ocean experiments have shown that the ITCZ
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remains very close to the equator throughout the seasonal cycle for ocean depths on the order of
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100m (Donohoe et al. 2014; Bordoni and Schneider 2008). However, the ITCZ’s behavior for
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smaller ocean depths is less clear.
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Fig. 2 shows the zonally averaged precipitation over the seasonal cycle for three cases of varying
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heat capacities (see Table 1 for heat capacity values, slab ocean depth values, and thermal inertia
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timescales), as well as over the final year of the “eternal solstice” simulation. There is little change
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in the position of the ITCZ as the heat capacity decreases to drastically small values. This suggests
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that while the heat capacity affects the migration of the ITCZ at larger values, there is a lower limit,
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namely the control value, under which the heat capacity no longer affects the ITCZ’s migration. In
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other words, it appears that when the slab ocean depth is at the control value we reach a limit where
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the associated timescale for the energetic adjustment to the lower boundary is so small compared
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to the timescale of the seasonal cycle of insolation that further reductions in energetic adjustment
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timescale do not make any difference.
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We have focused on the surface heat capacity, but our “eternal solstice” simulation also addresses
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the effect of the atmospheric heat capacity on the ITCZ’s movement. Maximum insolation in
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the “eternal solstice” case is fixed at the pole for ten years, much longer than the timescale for
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atmospheric adjustments, and still the precipitation band does not migrate towards the summer
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pole. Thus, it appears that reducing the timescale for atmospheric adjustments even further would
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not have any impact on the ITCZ migration, as was the case for surface temperature adjustments.
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Together, the null results of reducing both surface and atmospheric heat capacities suggest that
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dynamical constraints, as related to rotation rate or radius for instance, may be operating in cases
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of low thermal inertia. The role of rotation in “forcing” the ITCZ towards the equator has been
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similarly put forth by Chao (2000) and Chao and Chen (2001), wherein two “forces” critical to
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monsoon onset are described as in balance: one, associated with Earth’s rotation, towards the equa-
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tor, and the other towards the SST peak. The extent to which our findings relate to these “forces”
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is unclear, however, since we employ a markedly different experimental design and analyze the
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ITCZ behavior through the perspective of the boundary layer momentum budget rather than in the
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“forcing” framework of those studies.
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The ITCZ’s limited response to reductions in heat capacity also suggests that on Titan and Mars
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other planetary parameters other than the surface heat capacities may be the primary controls on
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the excursions of their seasonal convergence zones. Indeed, the atmospheric heat capacity on Titan
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is actually quite large (cf. Mitchell et al. 2014), indicating that the dynamical effect of its smaller
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rotation rate and radius must dominate strongly over the large atmospheric heat capacity to enable
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convergence over the summer pole.
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b. Maximum MSE
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Theory for the location of the convergence zone, based on the assumptions of a moist adiabatic
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vertical thermodynamic profile in statistical equilibrium, i.e. convective quasi-equilibrium (CQE),
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and an angular momentum-conserving meridional circulation with a vertical boundary, argues that
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the maximum zonal mean precipitation occurs just equatorward of the latitude of maximum zonal
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mean subcloud MSE (Emanuel et al. 1994; Privé and Plumb 2007). The low-level zonal mean
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MSE [m] = c p [T ] + [F] + Lv [q], where the brackets indicate zonal mean, is strongly correlated
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with surface temperature over a saturated surface, as in our aquaplanet simulations. Examining
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the ITCZ in the “eternal solstice” experiment provides further insight into the CQE argument by
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effectively eliminating energetic adjustment timescales, as noted in the previous section.
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Fig. 3 shows the zonal and time-mean precipitation, circulation features, and low-level MSE for
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the “eternal solstice” case. Maximum MSE in this case is located at the summer pole. The ITCZ,
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though, remains at low latitudes. The maximum zonal and time-mean precipitation occurs at
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approximately the same latitude for both the seasonal and “eternal solstice” case, around 20-25N.
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c. Discussion of thermodynamic mechanisms
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Neither heat capacity nor MSE exert a dominant control on the ITCZ’s poleward excursion,
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suggesting dynamics may be primarily driving the flow rather than thermodynamics in our sim-
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ulations. From Fig. 3, MSE gradients, rather than the MSE itself, essentially maximize at the
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latitude of maximum precipitation for both the seasonal and “eternal solstice” cases. This would
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seem to be consistent with the arguments of Plumb and Hou (1992) and Emanuel (1995), who
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demonstrate that in a moist convective atmosphere, a cross-equatorial Hadley circulation needs
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to exist (in place of a radiative equilibrium response) when the curvature of the subcloud moist
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entropy (or nearly equivalently MSE) exceeds a critical value. This theory therefore more pre-
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cisely defines the location of an overturning circulation by suggesting a link between the extent of
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the cross-equatorial Hadley cell, and with it the ITCZ, and the low-level MSE curvature. Please
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however note that the criticality condition can only be applied to radiative-convective equilibrium
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profiles of the low-level MSE: once an overturning develops when this critical condition is met,
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the circulation itself will determine the MSE distribution. Hence, this criticality argument cannot
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be easily translated into a quantitative diagnostic of the ITCZ position for our simulations. We
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do confirm that in an eternal solstice axisymmetric simulation run under our control parameters,
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the extratropical moist entropy distribution is indeed subcritical with regards to the critical con-
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dition calculated via a radiative-convective perpetual solstice experiment (not shown). Thus, at
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least for that case, though the zonal MSE maximizes at the pole, the extratropical atmosphere is in
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radiative-convective equilibrium and there is no circulation at those latitudes, consistent with the
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arguments just described.
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As we discuss in section 6, the simulated cross-equatorial circulations for all rotation rates ap-
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proach the angular momentum-conserving solution (Held and Hou 1980; Caballero et al. 2008),
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in which the potential temperature in the free troposphere adjusts through thermal wind balance
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to the angular momentum-conserving winds. Together with energy conservation constraints, these
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angular momentum-conserving arguments can indeed be used to determine the latitude of the as-
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cending branch of the winter Hadley cell, which, while not equivalent to, correlates well with the
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ITCZ position. The strong correlation in Fig. 3 between the ITCZ and the maximum MSE gradient
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in both seasonal and “eternal solstice” cases is indicative of the importance of surface tempera-
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ture gradients in determining convergence, arguing for the role of BL momentum dynamics, in
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agreement with Lindzen and Nigam (1987) and Back and Bretherton (2009).
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4. Baroclinic instability
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The “eternal solstice” case presented in the previous section also addresses the impact of baro-
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clinic instability on the ITCZ. It has been suggested that baroclinic eddies may limit the extent of
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the Hadley circulation (Schneider 2006; Levine and Schneider 2015). Observational studies have
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linked the contraction and expansion of the Hadley cell during El Niño/La Niña events to anoma13
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lous meridional temperature gradients and subtropical baroclinicity (Lu et al. 2008; Nguyen et al.
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2013). Since the summer cell is very weak compared to the winter cell during solstice, it is pos-
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sible that baroclinic eddies might limit the excursion of the cross-equatorial winter cell into the
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summer hemisphere. Additionally, eddy momentum fluxes have been shown to affect the large-
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scale meridional circulation in a variety of idealized climates (Walker and Schneider 2006; Levine
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and Schneider 2011), and to play a large role in monsoonal transitions in solstitial circulations
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(Bordoni and Schneider 2008; Schneider and Bordoni 2008; Merlis et al. 2013). During peak sol-
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stice, eddy momentum fluxes are relatively weak in the region of ascent and the cross-equatorial
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winter circulation is nearly angular momentum-conserving, shielded from eddies by upper-level
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easterlies; but on the poleward flank of the winter cell in the summer hemisphere, eddy momen-
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tum fluxes, though weaker than in the winter hemisphere, largely balance the westward Coriolis
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force from the equatorward flow in the upper branch. It remains unclear whether this extratropical
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baroclinicity is responsible for limiting the migration of the cross-equatorial winter Hadley cell,
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and therefore the ITCZ, into the summer hemisphere.
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From Fig. 3, in the seasonal case poleward of the MSE maximum, there is a small gradient,
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suggesting baroclinic eddies may be present and preventing the winter Hadley cell from extending
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farther polewards. However, in the “eternal solstice” case, no such gradient exists, indicating a
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complete lack of baroclinic activity in this region, with zonal MSE maximizing at the pole. Yet,
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the circulation does not become global and its poleward extent remains at low latitudes. These re-
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sults, therefore, suggest that extratropical baroclinic instability plays no role in limiting the ITCZ’s
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poleward migration during solstice in our simulations. Furthermore, in a simulation with the same
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seasonal cycle and Earth’s rotation rate run under axisymmetric conditions—in which the forma-
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tion of baroclinic eddies is entirely suppressed—the ITCZ and Hadley cell extent still remain at
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low latitudes (not shown).
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Given that neither baroclinic instability nor the thermodynamic constraints of heat capacity and
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MSE maxima seem to control the ITCZ in our simulations, in the next sections we explore possible
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dynamical controls related to the BL momentum budget.
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5. Rotation rate experiments
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To explore the ITCZ through a more dynamical perspective, we run Earth-like simulations for
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varying rotation rates. This is roughly equivalent to varying radius since both parameters similarly
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impact Hadley cell width (Held and Hou 1980). We find the ITCZ, defined here as the latitude of
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maximum zonally and solstitially averaged precipitation, reaches the summer pole for W/WE = 18 .
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a. Precipitation
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Fig. 4 shows zonal mean precipitation for several rotation rate cases. The precipitation band of
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the ITCZ moves farther off the equator towards the summer pole with each decrease in rotation
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rate, consistent with previous studies.
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It is worth noting that in the quickly rotating cases, including the control case, there are small
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“patches” of polar precipitation separated from the main ITCZ band at lower latitudes. This local
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precipitation is not strong enough for the ITCZ as we’ve defined it to be located at the pole, but it
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is consistent with the fact that local maxima in MSE exist at the pole during solstice in all cases,
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as shown in Fig. 5. Since low-level humidity is essentially uniform in an aquaplanet, local SST
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correlates extremely well with MSE. Thus, polar maxima in MSE during solstice reinforce the
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argument that for some planetary regimes, the ITCZ does not simply follow maximum zonal SST
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or MSE. Please note that though solstitial averages were taken prior to these polar maxima in the
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quickly rotating cases (see vertical orange lines in Fig. 5), the zonal circulation remains at lower
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latitudes throughout the summer, as indicated by the main ITCZ band.
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We also run simulations at slower rotation rates, down to 32 times smaller than Earth’s, and find
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the ITCZ reaching the summer pole in those cases as well, but they are not shown because their
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circulations are similar in structure to that of the W/WE =
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the planet enters a regime where the winter Hadley cell becomes global with the ITCZ at the pole,
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precluding any farther migration with larger decreases in rotation. Similarly, the W/WE = 2 and
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W/WE = 4 cases are not shown due to their circulation structures largely resembling that of the
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control case, though their ITCZs are indeed closer to the equator than the control (see Fig. 8).
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b. Circulation
1
8
case. After W/WE = 18 , it appears
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The ITCZ is closely associated with the ascending branch of the Hadley circulation. The right
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columns of Fig. 6 and Fig. 7 show the zonally averaged meridional circulation and angular
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momentum contours during solstice for all rotation cases. Consistent with the precipitation, the
343
cross-equatorial Hadley cell expands farther poleward for each decrease in rotation rate. Addi-
344
tionally, the summer cell is negligible when compared to the cross-equatorial winter cell for all
345
simulations during the averaged solstitial time period.
346
In aquaplanet simulations of Earth with negligible surface thermal inertia, as the seasonal cy-
347
cle transitions from equinox to solstice—representing monsoon onset over land-dominated re-
348
gions such as the Asian monsoon region—the winter Hadley cell strengthens and its ascending
349
upper branch becomes more angular momentum-conserving (Lindzen and Hou 1988; Bordoni
350
and Schneider 2008; Schneider and Bordoni 2008). At solstice, upper-level easterlies shield the
351
circulation from energy-containing midlatitude eddies and the eddy momentum flux divergence is
352
weak, thus allowing streamlines to follow angular momentum contours in the ascending branch
353
of the cross-equatorial winter cell. Poleward of this ascending branch, where the summer cell is
354
significantly diminished and meridional temperature gradients are larger, eddy momentum fluxes
16
355
balance the Coriolis force at upper levels. We observe similar solstitial dynamics as the rotation
356
rate is decreased in our simulations: streamlines in the ascending region of all cases generally
357
follow angular momentum contours; and in all cases a region of upper-level easterlies (not shown)
358
is maintained by the winter circulation, which continues to expand latitudinally with decreasing
359
rotation rate down to W/WE = 18 , where the circulation finally becomes global.
360
Thus, a regime change occurs after W/WE = 18 , wherein three conditions hold for planets with
361
that rotation rate and slower: 1) summer precipitation reaches the pole, 2) the winter cross-
362
equatorial cell extends from pole to pole and has a latitudinally wide region of updraft as op-
363
posed to the meridionally narrow ascent region seen in the more quickly rotating cases, and 3)
364
eddy momentum flux divergence is weak at all latitudes. A similar regime change was noted in
365
the non-seasonal experiments of Walker and Schneider (2006), wherein the upper branches of the
366
circulations for slowly rotating planets were nearly angular momentum-conserving around the lati-
367
tude of the Hadley streamfunction extremum. However, as noted in the previous section, baroclinic
368
instability appears to play no role in limiting the migration of the ITCZ.
369
c. Diagnostics for the ITCZ
370
Here we evaluate relevant diagnostics for the ITCZ, namely the Hadley cell extent, the maximum
371
low-level MSE, and the energy flux equator. As in the previous section, the zonally averaged low-
372
level moist static energy [m] is taken at 850 hPa, with the overbar denoting solstitial average. The
373
Hadley cell extent fH is defined as the latitude where the cross-equatorial streamfunction, taken
374
at the pressure level of its maximum value, reaches 5% of that maximum value in the summer
375
hemisphere (Walker and Schneider 2006). Finally, the energy flux equator defined in Kang et al.
376
(2008) as where the energy flux F = h[mv]i reaches zero, with the angled brackets denoting vertical
377
integration. However, since F doesn’t vanish until the pole for W/WE =
17
1
2
and slower during
378
solstice, here we define the energy flux equator as the latitude where the flux F reaches 5% of
379
its maximum value, similar to the Hadley cell extent definition. Fig. 8 shows these diagnostics,
380
each normalized to the maximum zonally and solstitially averaged precipitation (the ITCZ), for all
381
rotation cases. It also shows the latitude where the function G = 0.5, with G defined in Eq. 4 in
382
section 6.3.
383
The MSE maximum, Hadley cell extent, and energy flux equator occur poleward of the latitude
384
of the ITCZ for all but the most slowly rotating case. These three values are generally well corre-
385
lated (particularly at faster rotation rates). Indeed, the Hadley cell extent and energy flux equator,
386
physically related to one another (and, by definition, nearly equivalent), are approximately col-
387
located for each rotation rate. It is also clear from Fig. 8 that the ITCZ is associated with the
388
ascending branch of the cross-equatorial winter Hadley cell but generally lies equatorward of its
389
edge. However, the distance of separation between the ITCZ and the winter Hadley cell extent
390
appears to increase with decreasing rotation rate, until the circulation enters the slowly rotating
391
regime. We also note the larger discrepancy between the latitude of maximum MSE and the ITCZ
392
latitude in the W/WE =
393
the convergence zone as rotation rate slows: The distinction between the main precipitation band
394
of the ITCZ and the polar precipitation becomes less clear and the circulation appears on the verge
395
of entering the slowly rotating regime of global circulation, as demonstrated by the W/WE =
396
case.
1
4
and W/WE =
1
6
cases. This is indicative of the increasing migration of
1
8
397
That the maximum precipitation is slightly equatorward of the maximum MSE during solstice
398
has been noted in both models and observations (Privé and Plumb 2007; Bordoni and Schneider
399
2008). Discrepancies between the energy flux equator and the ITCZ—in both directions, i.e.
400
with the energy flux equator both undershooting and overshooting the latitude of the ITCZ—have
401
also been observed in various models (Kang et al. 2008; Bischoff and Schneider 2014), further
18
402
emphasizing the need for a more robust predictor of the ITCZ extent, particularly in strongly
403
off-equatorial regimes. In the next section, we work towards developing an understanding of the
404
position of the ITCZ based on BL dynamics and TOA energy balance.
405
6. Evaluating dynamical mechanisms behind the ITCZ position
406
In this section, we analyze potential dynamic controls on the ITCZ position. Before delving
407
into the details, it might be useful to first give an overall qualitative picture of the mechanisms
408
at work in our simulations. The flow that leads to the ITCZ is part of the lower branch of the
409
cross-equatorial winter Hadley cell. Using the case of northern summer solstice as an example,
410
this flow is the northward BL flow between the equator and the northern edge of the winter cell
411
in the summer hemisphere. The edge of the winter cell itself is where the flow vanishes, at a
412
latitude analogous to the energy flux equator or the maximum MSE, and can be determined by
413
TOA energy balance (Held and Hou 1980; Lindzen and Hou 1988; Satoh 1994; Caballero et al.
414
2008). Meridional forces in the BL, primarily the northward pressure gradient force and southward
415
Coriolis force, govern the zonal mean flow. However, while eddy momentum flux divergence
416
fluxes and vertical advection are very small in the BL, nonlinear advective forces are not and so
417
the flow is not completely in geostrophic balance. The pressure gradient, Coriolis, and nonlinear
418
forces combine into a northward force that balances the southward drag. Since the flow must
419
vanish farther poleward at the Hadley cell edge—and indeed all the forces must vanish there,
420
except for in the slowly rotating regime where they cannot vanish before reaching the pole—there
421
is a flow transition that must occur that leads to the ITCZ.
422
423
Our simulations demonstrate this mechanism and also provide a scaling for the latitude of the
ITCZ’s poleward excursion with rotation rate.
19
424
a. TOA energy balance
425
The mechanism behind the ITCZ, as it appears from our analysis, is intimately tied to previous
426
work done by Held and Hou (1980) and in particular by Caballero et al. (2008), who applied the
427
energy balance arguments of Held and Hou (1980) to the solstitial circulation in order to determine
428
the extent of the cross-equatorial winter Hadley cell into the summer hemisphere. The connection
429
to the ITCZ lies in the idea that the Hadley cell edge as determined by these energy balance
430
arguments corresponds to the point where the energy flux and boundary layer forces vanish, just
431
poleward of the ITCZ.
432
In Held and Hou (1980), the Hadley cell extent given equatorially symmetric insolation is deter-
433
mined through two primary requirements: that the energy budget of the Hadley cell is closed, and
434
that the temperature structure at the cell’s poleward edge matches the radiative-convective equilib-
435
rium profile. These two requirements are equivalent to saying that the TOA radiative imbalance
436
integrated over the width of the Hadley cell is equal to zero, and that the TOA radiative imbalance
437
at the latitude of the cell’s poleward edge is itself zero. Note that in these theories the winds in
438
the upper branch of the Hadley cell are assumed to be angular momentum-conserving. With these
439
requirements, it is possible to determine the Hadley cell extent, which is dependent on the thermal
440
441
442
443
Rossby number such that fH µ (Ro)1/2 for Ro =
gHDH
,
W2 a2
where H is the tropopause height and DH
is the pole-to-equator radiative-convective equilibrium temperature difference.
Caballero et al. (2008) applied similar arguments to seasonal insolation and found that the solstitial Hadley cell extent followed a different power law:
fH µ (Ro)1/3
444
445
(1)
Fig. 9 shows the scalings of the ITCZ and the energy flux equator with rotation rate. Cases
slower than W/WE =
1
6
were not included in the scaling (W/WE =
20
1
8
is in the plot, but not the
446
scaling calculation) because the ITCZ’s progression with rotation rate is limited by the pole beyond
447
that point: once the Hadley cell is global, it remains so for slower rotation rates. The W/WE = 2
448
and W/WE = 4 cases, however, with ITCZs correspondingly close to the equator, were included in
449
the scaling.
450
Since Ro scales as W 2 , the solstitial Hadley extent as determined by Caballero et al. (2008)
2/3 ,
451
scales as W
452
equator and the latitude of the ITCZ itself—despite the fact that Caballero et al. (2008) used a
453
dry axisymmetric model. This may be associated with the dynamic characteristics of solstice in an
454
aquaplanet simulation: during solstice, the upper branch of the winter Hadley cell is approximately
455
angular momentum-conserving at the latitude of ascent, with local Rossby numbers Ro
456
signifying the relatively weak impact of eddy momentum flux divergence (Bordoni and Schneider
457
2008; Schneider and Bordoni 2008), thereby approximating axisymmetric conditions. As a caveat
458
to the similarities just described, we should note that the power laws of Held and Hou (1980) and
459
Caballero et al. (2008) employ the small-angle approximation, which is most likely inappropriate
460
for our most slowly rotating cases.
and Fig. 9 shows a similar scaling in our simulations—both for the energy flux
0.6,
461
Plotting the insolation against the outgoing longwave radiation (OLR) visually illustrates these
462
energy balance arguments. Given the second requirement of the theory, the Hadley cell edge is
463
where the radiative imbalance is zero, i.e. where the insolation and OLR intersect. But given the
464
first requirement, the budget is closed and so the integrated imbalance over the width of the Hadley
465
cell is zero, meaning the Hadley cell is determined by an equal-area construction of the imbalance.
466
Fig. 10 shows the insolation and OLR for the end-member rotation rates, with and without
467
eddies, as well as with and without “eternal solstice” forcing. The figure illustrates that the ra-
468
diative structure of the deep tropics looks very similar across all parameters and indeed gives the
469
approximate width of the Hadley cell. Though large-scale eddies clearly impact the circulation
21
470
(particularly at higher latitudes) and prevent the insolation and OLR curves from intersecting at
471
the Hadley cell edges in most cases, the winter Hadley cell edge in the summer hemisphere corre-
472
sponds roughly to the local maxima of OLR.
473
Thus, the Hadley cell extent can be approximately determined by the shape of the OLR curve.
474
In the axisymmetric “eternal solstice” case at Earth’s rotation rate, for example, the summer hemi-
475
sphere edge of the winter cross-equatorial Hadley cell is at v30N, coinciding with the local max-
476
imum in OLR there. The Hadley cell extents plotted in Fig. 10 are defined as the latitudes where
477
the cross-equatorial streamfunction, taken at the pressure level of its maximum value, reaches 2%
478
of that maximum value, rather than when it vanishes entirely. This could account for the discrep-
479
ancies between the insolation and OLR in the winter hemisphere, where the region of descent is
480
relatively broad in latitude as compared to the narrow region of ascent in the summer hemisphere;
481
but winter hemisphere eddies may also be contributing to the discrepancies in the eddy-permitting
482
cases. The axisymmetric “eternal solstice” case, the most steady and without eddies, gives the
483
clearest picture of the energy balance arguments, with the equal-area construction almost exactly
484
matching the winter Hadley cell width. However, the same structure is also apparent in the control
485
seasonal case: even though the insolation and OLR don’t quite intersect, the OLR profile flattens
486
near the edges of the winter Hadley cell.
487
For the W/WE =
1
8
cases, on the other hand, the equal-area argument suggests the cell is global.
488
In other words, the edges of the winter cell—in loose terms, since the OLR never actually intersects
489
the insolation at either edge of the cell in any case—are essentially at the poles. In some sense, the
490
cell’s edge must be at the pole because it cannot be any farther poleward than that.
22
491
b. Boundary layer dynamics
492
Our analysis of the BL dynamics will be conducted primarily through the lens of the momentum
493
equations as presented by Schneider and Bordoni (2008), hereafter SB08, who studied Earth’s
494
solstitial dynamics in the convergence zone using a dry GCM with a seasonal cycle. Following
495
SB08, the BL steady-state momentum equations with drag represented as Rayleigh drag with
496
damping coefficient e are:
497
( f + z )v ⇡ eu
( f + z )u +
∂B
⇡ ev
∂y
(2)
(3)
498
where B = F + (u2 + v2 )/2 is the mean barotropic Bernoulli function with geopotential F and the
499
overbar again denotes the time-mean. Formulating the momentum equation this way means that
500
the Bernoulli gradient accounts for the pressure gradient force as well as nonlinear contributions.
501
Aside from approximating turbulent drag as Rayleigh drag, the other assumptions made here are
502
that contributions from eddy momentum flux divergence and vertical advection are negligible, and
503
indeed in our simulations those terms are about an order of magnitude smaller than the retained
504
Coriolis, Bernoulli gradient, drag, and z u terms (We include the Rayleigh drag approximation
505
here because it will be used in the next subsection, but the drag force plotted in Fig. 6 and Fig. 7
506
is diagnosed directly from the model, calculated by horizontal diffusion).
507
Fig. 6 and Fig. 7 give a summary of these forces for each rotation case during northern summer
508
solstice and the “eternal solstice” case, with all values vertically integrated over the BL. We define
509
the BL as ending at 850 hPa, though for the more slowly rotating cases, the lower branch of the
510
circulation reaches the free troposphere. For all cases, the ITCZ is well correlated with the latitude
511
of maximum BL drag. Indeed, in the “eternal solstice” case, where the maximum MSE condition
23
512
failed, the ITCZ and latitude of maximum drag are exactly collocated. From the right column of
513
Fig. 6 and Fig. 7, it is evident that for each case the ITCZ is located in the BL region of maximum
514
southward drag, a region just equatorward of the edge of the winter Hadley cell.
515
This mechanism seems to break down for the most slowly rotating case, with the latitude of
516
maximum drag lying far equatorward of the ITCZ. This may be due to the widening of the con-
517
vergence zone but could also be related to the change in the convergence zone force balance from
518
primarily geostrophic in the quickly rotating cases to something closer to cyclostrophic in the
519
slowly rotating regime, with the nonlinear z u force becoming a dominant contributor to balancing
520
the Bernoulli gradient force. Also in the slowly rotating cases, regions of strong meridional drag
521
exist at equatorial latitudes. Though they do not appear in the mass flux contours, there are indeed
522
secondary maxima of precipitation and vertical velocity in these regions. The mechanism for these
523
secondary equatorial convergences is left for future work, but it is perhaps similar to that described
524
in Pauluis (2004), where the cross-equatorial jump is due to a weak SST gradient.
525
It is also apparent from the left column of Fig. 6 and Fig. 7 that the ITCZ is associated with
526
the maximum northward Bernoulli gradient force, consistent with the previous observation of the
527
simulated ITCZ being well correlated with the maximum MSE gradient, given that temperature
528
gradients determine pressure gradients. In many cases, in fact, the ITCZ is collocated with the
529
maximum northward geopotential gradient, which lies just equatorward of the maximum Bernoulli
530
gradient. This is notable in that it marks the importance of temperature (and therefore pressure)
531
gradients in driving the convergence zone flow (Lindzen and Nigam 1987; Tomas and Webster
532
1997; Pauluis 2004; Back and Bretherton 2009). However, due to the increasing relevance of
533
nonlinear advective forces (namely the z u force) as the ITCZ moves farther from the equator in
534
the more slowly rotating cases, the linear theories of Lindzen and Nigam (1987) and Back and
24
535
Bretherton (2009) cannot be applied here, as they predict an ITCZ too far poleward in all but our
536
control simulation.
537
Indeed, the significance of the z u force implies that classic Ekman theory arguments describing
538
flow—wherein the balance of forces is primarily between the pressure gradient force, the Cori-
539
olis force, and turbulent drag—are inappropriate when considering solstitial convergence zone
540
dynamics. According to Ekman theory, uplift maximizes where geostrophic vorticity maximizes,
541
but in every one of our simulations geostrophic vorticity maximizes well poleward of the ITCZ
542
during solstice (not shown), underlining the unsuitability of Ekman theory in solstitial situations
543
and the importance of nonlinearity in determining the BL maximum convergence—a relationship
544
also noted by Tomas et al. (1999).
545
c. Connection to SB08
546
We have just shown that the maximum meridional drag and maximum Bernoulli gradient ad-
547
equately describe the ITCZ in a diagnostic sense. Here we take steps towards attaining a more
548
predictive theory for the ITCZ’s location. The convergence zone momentum budget analysis in
549
SB08 begins with equations (1) and (2) and arrives at expressions for the zonal and meridional
550
winds in terms of a nondimensional function G, defined as:
G=
e2
e 2 + ( f + [z ])2
(4)
551
This function helps describe the solstitial convergence zone. For G ! 1, as seen from expression
552
(2), the BL balance is primarily between the Bernoulli gradient force and friction. As G ! 0, the
553
balance is primarily between the Bernoulli gradient force and the Coriolis force.
554
The connection to the BL convergence zone is that, during solstice, the zonal and time-mean
555
G profile transitions meridionally from values v1 near the equator, to values closer to 0 farther
25
556
557
poleward in the region of ascent at the edge of the cross-equatorial winter Hadley cell. Since
[v] ⇡
G ∂ [B]
e ∂y ,
this transition occurs in the convergence zone, equatorward of the Hadley cell edge.
558
Such a relationship led SB08 to approximate the region of greatest convergence, or the ITCZ, as
559
the region where G ⇡ 0.5. We observe a similar transition in our simulations, though note that the
560
latitude where G ⇡ 0.5 does not correlate well with the ITCZ position in any case (see Fig. 8).
2
561
562
2
Fig. 6 and Fig. 7 show the G profiles for each rotation case, where e = Cd ([u] + [v] )1/2 /H0 , with
Cd being the surface drag coefficient of momentum and H0 being the height of the BL (Lindzen
563
and Nigam, 1987). Note that e is not a constant but is dependent on the surface winds, since the
564
drag force reduces to the surface stress when vertically integrated in the BL.
565
In each case, the ITCZ occurs as G decreases to zero, in most cases at a value closer to G ⇡ 0.1
566
0.2. It remains unclear how to apply the theory underlying the G function to an exact prediction or
567
scaling of the ITCZ latitude. The ITCZ appears to correlate with the curvature of the G function,
568
but we have not developed a theoretical basis for such a relationship. The transition occurs over
569
a larger latitudinal range as rotation rate decreases, consistent with the widening winter Hadley
570
cell and more extensive regions of convergence and precipitation in those cases. It is also notable
571
that at the latitude where G ⇡ 1, the zonal and time-mean absolute vorticity [h] is approximately
572
zero in every case (not shown). This is consistent with the work of Tomas and Webster (1997),
573
in which the [h] = 0 latitude demarcates between divergence and convergence, with divergence
574
equatorward of that latitude and convergence associated with inertial instability—producing the
575
ITCZ—poleward of that latitude.
576
Finally, Fig. 6 and Fig. 7 show that the same approximate force balance holds for each case. One
577
noteworthy difference however is that in the “eternal solstice” case, the extratropical geostrophic
578
force balance is in the opposite direction than it is in the seasonal case. This can be attributed
579
to thermal wind balance and the fact that MSE maximizes at the pole in the “eternal solstice”
26
580
case, resulting in radiative equilibrium in the extratropics wherein there is no circulation, as noted
581
previously. Thus, the temperature profile controls the winds, producing high-latitude easterlies
582
and a northward Coriolis force. The unique radiative character of the “eternal solstice” case may
583
also be responsible for the stronger circulation and wider Hadley cell width than observed in the
584
seasonal case, though the ITCZ itself remains at the same latitude.
585
7. Summary and Conclusions
586
We conduct a suite of experiments to test various controls on the seasonal excursions of the ITCZ
587
using an idealized, aquaplanet Earth GCM. These controls include heat capacity, low-level moist
588
static energy, baroclinic instability, atmospheric energy balance, and boundary layer dynamics.
589
We find that decreasing the slab ocean depth beyond 10m does little to extend the ITCZ. We also
590
find, through “eternal solstice” experiments, that the ITCZ is not always closely coupled with the
591
maximum MSE and is also not limited by baroclinic instability: in the “eternal solstice” case, the
592
maximum MSE is located at the pole and there is little to no baroclinic instability in the summer
593
hemisphere, yet the ITCZ remains at low latitudes—exactly where it sits in the control Earth
594
simulation.
595
We slow down the rotation rate of an otherwise Earth-like planet to examine the ITCZ as it
596
migrates farther off the equator. We find the ITCZ reaches the summer pole for rotation rates
597
W/WE =
598
and energy flux equator, are generally collocated with each other, but robustly lie poleward of the
599
latitude of the ITCZ, with the distance of separation increasing as the ITCZ moves farther from
600
the equator. Based on the ITCZ’s behavior across a range of rotation rates, we describe a frame-
601
work for understanding the location of the ITCZ based on energy balance and BL dynamics—the
602
latter being very similar in principle to work previously done by Schneider and Bordoni (2008).
1
8
and smaller. Classic ITCZ diagnostics, namely maximum MSE, Hadley cell extent,
27
603
TOA energy balance, based on the axisymmetric theory of Held and Hou (1980), determines the
604
latitude poleward of the ITCZ where the flow vanishes. The ITCZ diagnostics mentioned above
605
correlate with this latitude rather than the ITCZ itself, which is the latitude of uplift and conver-
606
gence equatorward of the Hadley cell edge and closely associated with maximum BL drag and
607
Bernoulli gradient.
608
Common throughout our investigation has been the importance of temperature gradients and by
609
association pressure gradients. In the “eternal solstice” case where the maximum MSE condition
610
fails, the ITCZ is approximately collocated with the maximum MSE gradient; and in all cases,
611
the ITCZ is closely correlated with the maximum Bernoulli gradient force. This is in accord
612
with dynamical studies of the past, particularly Lindzen and Nigam (1987), Back and Bretherton
613
(2009), and Tomas and Webster (1997).
614
Such relationships are important when thinking about analogous systems of seasonal conver-
615
gence on other planets. Our results suggest that it is primarily the planetary parameters associated
616
with the angular momentum budget, i.e. the rotation rate and/or radius, that are responsible for
617
Titan’s and Mars’s far-reaching convergence zones, rather than their low surface heat capacities.
618
Indeed, the atmospheric circulations of the W/WE =
619
precipitation and a global Hadley cell, are not too dissimilar from that of Titan, which itself has a
620
rotation rate approximately one sixteenth of Earth’s and a radius one third of Earth’s. While we
621
are not yet able to predict the poleward migration of the ITCZ given only planetary parameters,
622
the axisymmetric scaling of Caballero et al. (2008) seems to be quite accurate when applied to our
623
simulations, even though they are moist and eddy-permitting. It is left for future work to further
624
develop our diagnostic ITCZ understanding into a more prognostic theory that can be applied to
625
any terrestrial planet with seasonally migrating convergence zones.
28
1
8
case and slower, which have intense polar
626
Acknowledgments. Simulations were carried out using the Hoffman2 Cluster hosted by UCLA’s
627
Institute for Digital Research and Education. The comments of three anonymous reviewers helped
628
greatly strengthen the final manuscript. S.B. acknowledges support by the National Science Foun-
629
dation Grant AGS-1462544. S.F. and J.M. were supported by NASA grants NNX12AI71G and
630
NNX12AM81G, and S.F. was supported in part by UCLA’s Eugene V. Cota-Robles Fellowship.
631
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765
LIST OF TABLES
766
Table 1.
767
Equivalent values of relevant parameters for heat capacity experiments (with
Earth included for reference) . . . . . . . . . . . . . .
36
.
37
768
769
TABLE 1. Equivalent values of relevant parameters for heat capacity experiments (with Earth included for
reference)
Heat capacity [J m
2
K
1]
Slab ocean depth
Thermal inertia timescale
Earth
5.0 ⇥107
50 m
110 days
Control simulation
1.0 ⇥107
10 m
22 days
Heat capacity simulation A
1.0 ⇥105
10 cm
5.3 hours
Heat capacity simulation B
1.0 ⇥103
1 mm
3.2 mins
37
770
LIST OF FIGURES
771
Fig. 1.
Left) Seasonal insolation forcing. Right) “Eternal solstice” forcing. Contours are in W/m2 . .
. 40
772
Fig. 2.
Zonal mean precipitation (color contours, mm/day) in tenth year of control simulation with
heat capacity CE = 107 J m 2 K 1 (top left), heat capacity experiment A (top right), heat
capacity experiment B (bottom left), and “eternal solstice” simulation with heat capacity
CE (bottom right). All four simulations were run at Earth’s rotation rate. See Table 1 for
equivalent slab ocean depths and thermal inertia timescales. . . . . . . . . .
. 41
773
774
775
776
777
Fig. 3.
778
779
780
781
782
783
784
785
Fig. 4.
786
787
788
789
Fig. 5.
790
791
792
Fig. 6.
793
794
795
796
797
798
799
800
Left) Zonal and time-mean low-level (850 hPa) MSE (green) and precipitation (blue) for
“eternal solstice” case at Earth’s rotation rate (dashed) and averaged over northern summer solstice for control seasonal case (solid). The green circle and diamond represent the
latitudes of the maximum MSE gradient for the seasonal and “eternal solstice” cases, respectively; Right) Zonal and time-mean circulation for “eternal solstice” case at Earth’s
rotation rate. Streamfunction contours (black, solid lines counter-clockwise) are 10% of
maximum, printed in top left corner in 109 kg/s. Grey contours are angular momentum
contours (Wa2 /17 interval).
. . . . . . . . . . . . . . . . .
.
42
Zonal mean precipitation for five rotation cases using same contour scaling as Fig. 2. Orange
vertical lines indicate time over which solstitial time averages are taken and represent the 20day period between ten days before the time of maximum zonal mean northern hemisphere
precipitation (black cross) and ten days after that time. . . . . . . . . . .
.
43
Zonal mean low-level MSE (color contours, kJ/kg) for five rotation cases. Orange vertical
lines indicate same time period as labeled in Fig. 3. Black cross indicates maximum zonal
mean northern hemisphere low-level MSE. . . . . . . . . . . . . . .
. 44
Left) Summer hemisphere forces on left y-axis, with positive values denoting a northward
force and where BERN, COR, and DRAG correspond to the Bernoulli gradient, Coriolis, and
frictional forces. Each force is diagnosed directly from simulation. Values of G, from Eq. 4,
are on right y-axis. Vertical blue line represents the latitude of maximum P. Vertical integration taken from the surface to s = 0.85. Solstitial time averages taken over same period as
previous figures. Right) Streamfunction, angular momentum, and drag. Streamfunction and
angular momentum contours same as in Fig. 3. Drag force (color contours) in m/s2 . Vertical
white line represents latitude of maximum vertically integrated BL drag force. Vertical blue
line represents latitude of maximum P as in left column. . . . . . . . . . .
.
45
801
Fig. 7.
Same as Fig. 6, but for more slowly rotating cases.
.
.
46
802
Fig. 8.
Maximum low-level MSE (green), Hadley cell extent (black), energy flux equator (red), and
the latitude where hGi ⇡ 0.5 (orange) as compared to the ITCZ (blue) for different rotation
cases. Blue line is 1:1. From left to right, rotation rates W/WE are: 4, 2, 1, 12 , 14 , 16 , 18 . . .
.
47
Rotation rate versus latitudes of ITCZ (blue) and energy flux equator (red). Dashed lines
represent scaling fits to cases with rotation rate W/WE = 16 and greater for both values (color
corresponds to color of crosses). . . . . . . . . . . . . . . . .
.
48
803
804
805
806
807
808
809
810
811
812
Fig. 9.
.
.
.
.
.
.
.
.
.
.
Fig. 10. Solstitial insolation (black) and zonally and solstitially averaged OLR (red) for four different
simulations at Earth’s rotation rate (left) and at W/WE = 18 (right): the seasonal case (top),
the eddy-permitting “eternal solstice” case (second from top), the axisymmetric seasonal
case (second from bottom), and the axisymmetric “eternal solstice” case (bottom). Vertical
black lines indicate northern and southern extents of winter Hadley cell, defined here as the
38
813
814
latitudes where the streamfunction, taken at the pressure level of its maximum value, reaches
2% of that maximum value. . . . . . . . . . . . . . . . . . .
39
. 49
F IG . 1. Left) Seasonal insolation forcing. Right) “Eternal solstice” forcing. Contours are in W/m2 .
40
815
F IG . 2. Zonal mean precipitation (color contours, mm/day) in tenth year of control simulation with heat
816
capacity CE = 107 J m
2
817
(bottom left), and “eternal solstice” simulation with heat capacity CE (bottom right). All four simulations were
818
run at Earth’s rotation rate. See Table 1 for equivalent slab ocean depths and thermal inertia timescales.
K
1
(top left), heat capacity experiment A (top right), heat capacity experiment B
41
819
F IG . 3. Left) Zonal and time-mean low-level (850 hPa) MSE (green) and precipitation (blue) for “eternal
820
solstice” case at Earth’s rotation rate (dashed) and averaged over northern summer solstice for control seasonal
821
case (solid). The green circle and diamond represent the latitudes of the maximum MSE gradient for the seasonal
822
and “eternal solstice” cases, respectively; Right) Zonal and time-mean circulation for “eternal solstice” case at
823
Earth’s rotation rate. Streamfunction contours (black, solid lines counter-clockwise) are 10% of maximum,
824
printed in top left corner in 109 kg/s. Grey contours are angular momentum contours (Wa2 /17 interval).
42
825
F IG . 4. Zonal mean precipitation for five rotation cases using same contour scaling as Fig. 2. Orange vertical
826
lines indicate time over which solstitial time averages are taken and represent the 20-day period between ten
827
days before the time of maximum zonal mean northern hemisphere precipitation (black cross) and ten days after
828
that time.
43
829
F IG . 5. Zonal mean low-level MSE (color contours, kJ/kg) for five rotation cases. Orange vertical lines
830
indicate same time period as labeled in Fig. 4. Black cross indicates maximum zonal mean northern hemisphere
831
low-level MSE.
44
832
F IG . 6. Left) Summer hemisphere forces on left y-axis, with positive values denoting a northward force and
833
where BERN, COR, and DRAG correspond to the Bernoulli gradient, Coriolis, and frictional forces. Each force
834
is diagnosed directly from simulation. Values of G, from Eq. 4, are on right y-axis. Vertical blue line represents
835
the latitude of maximum P. Vertical integration taken from the surface to s = 0.85. Solstitial time averages taken
836
over same period as previous figures. Right) Streamfunction, angular momentum, and drag. Streamfunction and
837
angular momentum contours same as in Fig. 3. Drag force (color contours) in m/s2 . Vertical white line represents
838
latitude of maximum vertically integrated BL drag force. Vertical blue line represents latitude of maximum P as
839
in left column.
45
F IG . 7. Same as Fig. 6, but for more slowly rotating cases.
46
840
F IG . 8. Maximum low-level MSE (green), Hadley cell extent (black), energy flux equator (red), and the
841
latitude where hGi ⇡ 0.5 (orange) as compared to the ITCZ (blue) for different rotation cases. Blue line is 1:1.
842
From left to right, rotation rates W/WE are: 4, 2, 1, 12 , 14 , 16 , 18 .
47
843
F IG . 9. Rotation rate versus latitudes of ITCZ (blue) and energy flux equator (red). Dashed lines represent
844
scaling fits to cases with rotation rate W/WE =
845
crosses).
1
6
and greater for both values (color corresponds to color of
48
846
F IG . 10. Solstitial insolation (black) and zonally and solstitially averaged OLR (red) for four different simula1
8
847
tions at Earth’s rotation rate (left) and at W/WE =
848
solstice” case (second from top), the axisymmetric seasonal case (second from bottom), and the axisymmetric
849
“eternal solstice” case (bottom). Vertical black lines indicate northern and southern extents of winter Hadley
850
cell, defined here as the latitudes where the streamfunction, taken at the pressure level of its maximum value,
851
reaches 2% of that maximum value.
(right): the seasonal case (top), the eddy-permitting “eternal
49