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# Extratropical forcing and tropical rainfall distribution: energetics framework and ocean Ekman advection

• npj Climate and Atmospheric Sciencevolume 1, Article number: 2 (2018)
• doi:10.1038/s41612-017-0004-6
Revised:
Accepted:
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## Abstract

Intense tropical rainfall occurs in a narrow belt near the equator, called the inter-tropical convergence zone (ITCZ). In the past decade, the atmospheric energy budget has been used to explain changes in the zonal-mean ITCZ position. The energetics framework provides a mechanism for extratropics-to-tropics teleconnections, which have been postulated from paleoclimate records. In atmosphere models coupled with a motionless slab ocean, the ITCZ shifts toward the warmed hemisphere in order for the Hadley circulation to transport energy toward the colder hemisphere. However, recent studies using fully coupled models show that tropical rainfall can be rather insensitive to extratropical forcing when ocean dynamics is included. Here, we explore the effect of meridional Ekman heat advection while neglecting the upwelling effect on the ITCZ response to prescribed extratropical thermal forcing. The tropical component of Ekman advection is a negative feedback that partially compensates the prescribed forcing, whereas the extratropical component is a positive feedback that amplifies the prescribed forcing. Overall, the tropical negative feedback dominates over the extratropical positive feedback. Thus, including Ekman advection reduces the need for atmospheric energy transport, dampening the ITCZ response. We propose to build a hierarchy of ocean models to systematically explore the full dynamical response of the coupled climate system.

## Introduction

In the current climate of the Earth, 32% of the globally integrated annual rainfall falls within 10°S–10°N (only 17% of the surface area) in a narrow band of heavy precipitation, called the inter-tropical convergence zone (ITCZ). The ITCZ is so sharp meridionally that a small shift in its position can cause dramatic local variations in rainfall. For example, Kanton Island near the edge of the ITCZ receives the annual rainfall of only 3 mm day−1 in some years, while they can experience rainfall over 10 mm day−1 in other years due to subtle shifts of the ITCZ. Also, the ITCZ migrates seasonally, with its zonal-mean position varying from 8.3°S in austral summer to 7.9°N in boreal summer. For example, precipitation at Port Morseby in Papua New Guinea varies from 3 mm day−1 during dry months to 20 mm day−1 during wet months. As hundreds of millions of the world population depend critically on tropical rains for water and food, it is of foremost importance to understand the mechanism that controls the ITCZ position.

Although tropical rainfall traditionally was thought to be controlled by local tropical mechanisms,1 paleoclimatic evidence has pointed at the possibility of an extratropical influence on tropical climate. The paleoclimate records from the tropical regions exhibit striking resemblance to the abrupt climate changes over Greenland associated with the Dansgaard–Oeschger events.2,3 These paleoclimate data motivated the modeling studies of possible pathways for extratropics-to-tropics teleconnection by perturbing the extratropics in general circulation models. The extratropical perturbations include: increased ice cover in the high-latitude Northern Hemisphere (NH),4 a sustained addition of freshwater in the North Atlantic,5,6 an expansion of dark forests in the northern extratropics,7,8,9 changes in radiative forcing agents, such as black carbon, sulfate, and volcanic aerosols,10 ozone depletion over Antarctica,11 and enhanced Southern Ocean heat uptake.12 In all cases, the ITCZ is shifted toward the differentially warmed hemisphere or away from the differentially cooled hemisphere. These modeling studies indicate that the extratropical perturbations can cause a meridional shift of the ITCZ, supporting the paleoclimate evidence. In the past decade, significant progress has been made in understanding how ITCZ shifts are forced from the extratropics based on the atmospheric energy budget. Here, we briefly review major advances made in this area of research, and highlight potential importance of ocean dynamic adjustments in modulating the extratropical impact on tropical rainfall.

## ITCZ and atmospheric energy budget

Heavy rainfall in the ITCZ is caused by moisture convergence in the lower branch of the Hadley circulation (HC). The air masses that converge near the surface ascend and diverge in the upper troposphere. In general, the magnitude of dry static energy transport by the upper branch of the HC is larger than the magnitude of latent heat transport by the lower branch. Hence, atmospheric energy is transported in the same direction of the upper branch of the HC. Thus, the ITCZ closely corresponds to the latitude of maximum moisture convergence, and to the latitude where the atmospheric energy flux F A vanishes, which is called the energy flux equator $φ e$ 13,14 (refer to Fig. 1). Suppose the northern extratropics are heated relative to the southern extratropics. The inter-hemispheric energy contrast can be connected to the ITCZ shift through the zonal-mean and time-mean vertically integrated atmospheric energy balance

$∇⋅ F A =R-O+S$
(1)

where F A is the vertically integrated total meridional atmospheric energy transport (AET), R is the net downward radiative fluxes at the top-of-atmosphere (TOA), O is the ocean energy uptake due to transport and storage, and S is the specified thermal forcing into the atmospheric column either from the TOA or from the surface. Let δ denote the response to S. The AET F A can be expressed as $F A =-VΔm,$ where V is the HC mass transport by its lower branch (δV > 0 at the equator for a northward displaced ITCZ) and ∆m is the total gross moist stability (TGMS) that measures the efficiency at which the HC transports energy. The TGMS is usually positive, indicative of a larger energy transport by the upper branch than the lower branch of the HC.15,16

An energetics framework that relates the ITCZ response to the hemispherically differential heating was developed using experiments with atmospheric models coupled to a motionless slab ocean, in which energy can be transported within the atmosphere only (i.e., there is no change in O).13,15,17 Consider a simple case where cloud radiative effects are inhibited. The effect of extratropical forcing (S > 0 in NH and S < 0 in SH) reaches the subtropical edge of the HC via atmospheric eddy fluxes.15 The resultant meridional temperature gradients in the tropical upper troposphere cannot be sustained due to the smallness of the Coriolis parameter.18 In order to flatten temperature gradients, the HC responds by transporting excessive energy from the NH to the SH with a deficit of energy. In other words, the energy has to be transported southward across the energy flux equator in the reference state, which requires a northward shift of the energy flux equator. There are largely two ways to induce a northward shift of the energy flux equator, as demonstrated in Fig. 1. One is by lowering the y-intercept while keeping the overall meridional profile the same (i.e., δF A0 < 0, with the subscript 0 denoting the equator, and $∇⋅δ F A =0$). That is, an anomalously southward cross-equatorial AET leads to a northward shift of the energy flux equator.13,19 The other is by reducing the meridional slope of AET in the equatorial region with an unchanged y-intercept (i.e., $∇⋅δ F A <0$ and δF A0 = 0). That is, a reduction in the divergence of AET in the tropics, equivalent to the net energy input into the atmospheric column based on Eq. 1, leads to a northward shift of the energy flux equator.20,21 The former case is depicted in blue in Fig. 1 and the latter in red. The effect of extratropical forcing is primarily balanced by a cross-equatorial AET, while the effect of tropical forcing, such as El Niño primarily modulates the net energy input to the equatorial atmosphere.20

The response of the energy transport can be directly related to the response of mass transport in the lower troposphere, that is, δF A = −∆mδV, assuming a fixed Δm. Hence, the anomalous southward energy transport, which is accompanied by the northward shift of the energy flux equator, causes an anomalous northward moisture transport in the lower troposphere, corresponding to a northward ITCZ shift (Fig. 1). This elucidates how hemispherically asymmetric heating in the extratropics can affect tropical rainfall. Extratropical forcing tends to be amplified when cloud-radiative effects are included, whereas tropical forcing tends to be damped.13,15,22 Hence, remote extratropical forcing can actually be more effective at shifting the ITCZ than local tropical forcing.22 In summary, a thermal contrast between the hemispheres, introduced from the tropics or from the extratropics, can force an ITCZ shift in order to balance the atmospheric energy budget.

The energetics framework facilitates the development of a new theory for why the climatological zonal-mean ITCZ position is north of the equator, emphasizing the importance of the Atlantic meridional overturning circulation (AMOC) and its northward transport of energy across the equator.23,24,25 The effect of the AMOC is zonally homogenized by winds in the extratropics, thereby placing the ITCZ in the NH not only in the Atlantic but also in the Pacific.26 Traditionally, the ITCZ position was thought to be controlled by local mechanisms, such as the shape of coast lines1,27,28 and Andes topography.29 These local processes may be important for the zonal variations of the ITCZ26,30 as the global energetics framework applies to the zonal mean. The northward heat transport by the AMOC also explains why the NH is warmer than the SH on average.31 Seasonal variations in ITCZ location can also be thought of as the result of an anomalous HC that must transport energy toward the differentially cooled hemisphere.14,32 The energetics framework has been successfully adopted to explain the ITCZ shifts in response to a doubling of CO2 in comprehensive models33 and the response to aerosols in the 20th century.34 Moreover, the energetics framework has been applied to diagnose the double-ITCZ problem, which is a long-standing climate model bias.35 Cloud biases over the Southern Ocean, responsible for excessive warming over the SH,36 have been suggested to be one of the causes of the double-ITCZ.

## Limitations and ways forward

The energetics framework assumes that the TGMS, ∆m is constant. If ∆m can vary, $δ F A$ is no longer directly related to δV and cannot account for the ITCZ shift. For example, consider the limiting case with δF A = 0 where the present energetics framework predicts no ITCZ shift. If changes in ∆m were large, there would be substantial changes in the mass transport, despite δF A = 0, resulting in the ITCZ shift. One can consider the other limiting case where δF A ≠ 0 is accompanied by no ITCZ shift but results entirely from changes in ∆m (i.e., δF A = m). As δm becomes more dominant over δV in determining δF A, the energetics framework starts to fail in predicting the ITCZ response. The relative importance between δV and δm may depend on the type of forcing. An inter-hemispheric thermal contrast is expected to provoke a significant HC shift with relatively little changes in ∆m. Then, the energy flux equator can be used to infer the ITCZ shift. In contrast, a nearly uniform radiative forcing induces a rather subtle HC shift and changes in ∆m cannot be ignored.37,38,39,40 In this situation, the energy flux equator is no longer a good proxy for the ITCZ.16 Therefore, one must account for changes in the TGMS for a complete theory.

For the energetics framework to have a full predictive power, it must also predict how radiative feedbacks would rectify the external forcing. For example, the zonally asymmetric thermal forcing with zero zonal-mean component is expected not to cause any ITCZ shift assuming radiative feedbacks are linear. However, nonlinear cloud-radiative feedbacks can cause a significant shift in the zonal-mean ITCZ.41,42 Furthermore, a given inter-hemispheric forcing can cause an ITCZ shift that varies in magnitude by more than 100% when a convection scheme parameter is tuned in a model in a way that alters the strength of cloud radiative effect.13,43 Model spread in cloud response is a dominant source for model spread in the ITCZ shift in response to inter-hemispheric forcing.44 Hence, without properly understanding cloud radiative effects, the framework cannot make a useful quantitative prediction. Since tropical clouds are strongly tied to the circulation,45 one may seek to formulate a closed theory by constructing an empirical relationship between cloud radiative effect and circulation based on observations.

The energetics framework is based on the zonal-mean atmospheric energy budget, hence, has limited relevance to regional changes. The zonal-mean ITCZ lumps together rich regional structures, such as monsoons, one of the most prominent local features of tropical circulation. The zonal-mean ITCZ shift may not convey the proper information on the changes of regional rainfall pattern. For example, the zonal-mean ITCZ position shifts only by about 1° in latitude between El Niño and La Niña composites, while the east–west shifts are 22° in longitude.46 Also, during the early Holocene, compared to today, the zonal-mean ITCZ shift is estimated to be less than 1° latitude, while paleoclimate records infer that the regional ITCZ shifts reach up to 5° latitude.47 Although some aspects of regional features, such as the abrupt onset of monsoon are captured by the zonal-mean dynamics,48 planetary-scale waves are important. For example, the summer monsoons extend farther poleward in the eastern than western side of a major continent like the Afro-Eurasian.49,50,51 Thus, expanding the zonal-mean energetics framework to two horizontal dimensions is required for predictions of regional rainfall changes in the tropics, say by accounting for the zonal component of divergent AET.46,52

The application of the energetics framework is further complicated by ocean dynamical feedback. When the ocean circulation is allowed to respond (i.e., O is no longer fixed in Eq. 1), the partitioning of energy transport between the atmosphere and ocean will determine the magnitude of the ITCZ response. Recently, fully coupled models with a dynamical ocean are employed to examine the ITCZ response to high-latitude energy perturbation.53,54,55,56,57 In response to modified Southern Ocean albedo, the changes in cross-equatorial energy transport are dominated by ocean adjustments, and the ITCZ response is muted with a persistent double-ITCZ problem53,54 in some models (CESM-CAM558 and HadGEM2-ES59). Similarly, when the model (CCSM460) is forced by the projected Arctic sea ice loss, the climate response exhibits a high degree of hemispheric symmetry with little ITCZ shift, as the energy transport is mostly accomplished by the ocean in the equatorial region.61,62 In contrast, in a different model (GFDL AM4 coupled to FLOR63), the atmosphere and ocean nearly equally share the load of compensating the variations in the solar flux over the Southern Ocean, and a small but clear meridional ITCZ shift is identified.55 In another GFDL family of climate models (GFDL CM2Mc64), the changes in cross-equatorial energy transport equally occurred in the atmosphere and ocean in response to SH high-latitude radiative changes associated with Southern Ocean convective variability.57 In response to a reduction of solar flux over the Southern Ocean, the UCLA CGCM produces a significant ITCZ shift whereas a much weaker ITCZ response is obtained in the NorESM model.56 These studies that employ fully coupled models suggest a large inter-model spread in the degree by which the ocean circulation modulates the tropical response to extratropical forcing. In general, extratropical energy perturbations are not as effective at forcing the ITCZ shift as predicted by the energetics framework with a slab ocean. This is expected due to an intimate coupling between the subtropical ocean cell and the HC through the trade winds.65 As the ocean transports energy in the same direction as the HC, the amount of energy needed to be transported by the atmosphere will decrease, leading to a reduction in the tropical rainfall response. The cause of such a large inter-model spread in ocean dynamical compensation remains to be explored.

## Ocean dynamics and the ITCZ

This section considers ocean dynamical feedback that affects the partitioning of energy transport between the atmosphere and ocean. We attempt to provide a possible explanation for a range of sensitivity of tropical rainfall to extratropical forcing in coupled models with a fully dynamic ocean. In fully coupled models, it is difficult to disentangle the effect of different components of the large-scale ocean circulation including the thermohaline and wind-driven circulation, so for better understanding one might wish to reduce the complexity of the climate system.66 Therefore, we propose to build a hierarchy of climate models in which a simplified atmospheric model is coupled to ocean models of different levels of complexity. The atmospheric model we employ has no cloud radiative feedbacks, a key source of uncertainty, to focus on more fundamental interactions between latent heating and circulation.67,68 The ocean model is also intended to be as simple as possible yet contains dynamics of meridional energy transport. In the tropics, energy is transported poleward primarily by the wind-driven circulation, which consists of poleward Ekman flow at the surface and the equatorward return flow in the thermocline that upwells in the equatorial region.69,70,71,72 Here, we implement meridional Ekman advection into the aquaplanet slab ocean model, while neglecting the upwelling effect in the deep tropics and the subsiding effect in the subtropics.73 That is, we only incorporate the effect of meridional heat advection at the surface by the Ekman flow, without the need to include the vertical stratification of the ocean. The shallow overturning cell that sinks in the subtropics with the return flow in the thermocline will be considered in a subsequent study by adopting a two-layer ocean model.74 Systematically adding the complexity to the ocean model is expected to help develop fundamental understanding of how the ocean circulation modulates the extratropics-to-tropics teleconnection.

The meridional Ekman flow (v Ek) is induced by a zonal wind stress (τ x ):

$v E k =- f f 2 + ε 2 τ x ρ h$ where f is the Coriolis parameter, ε is the mechanical damping rate (5.7 × 10−6 s−1), ρ is the density of water (1 × 103 kg m−3), and h is the mixed layer depth (50 m) (see section “Methods” for more details). To simulate the effect of meridional Ekman heat advection, we add the term $-ρ C p h v E k ∂ T S a ∂ φ$ (where $∂ T S ∂ φ$ indicates meridional surface temperature gradients with the radius of the earth a and a specific heat at constant pressure C p) to the surface temperature tendency equation.75,76 The meridional energy transports by the Ekman flow F Ek are calculated across each latitude band (φ) by integrating the Ekman heating term from one pole to the other: $F E k φ = ∫ - π ∕ 2 φ ∫ 0 2 π - ρ C p h v E k ∂ T S a ∂ φ a 2 cosφdλdφ$, where λ is longitude. It is noted that our aquaplanet model may produce an upper bound of the zonally integrated Ekman mass transport and associated energy transport due to the absence of ocean basins. In regions with surface easterlies, the Ekman flow is directed poleward, leading to warm advection. Conversely, in regions with surface westerlies, the Ekman flow is directed equatorward, leading to cold advection. Hence, when Ekman advection is implemented, the equatorial peaks of the mean surface temperature and rainfall in the symmetric reference run are slightly enhanced, whereas the surface temperature in the mid-latitudes is slightly cooled (Fig. 3a and c). With the upwelling effect, which is to be considered in the subsequent study, the equatorial peak is expected to be damped, broadening the warm pool.

In both models with and without Ekman advection, the reference climate is simulated with no surface flux adjustments (S = 0). We then perturb the reference climate with an antisymmetric forcing S that extracts heat at the surface poleward of 40°S and injects it poleward of 40°N (Fig. 2a). Because of the zero global mean, the prescribed antisymmetric surface thermal forcing S can be described in terms of an implied northward ocean energy transport F S, with S = −·F S (Fig. 2b). The maximum amplitude of S is varied such that its corresponding cross-equatorial flux, denoted as F S0, ranges from 1.1 to 5.8 pW. The range of forcing would correspond to 0.28–1.45 pW in a comprehensive model where cloud radiative effects amplify the prescribed forcing by a factor of four.15

With the inclusion of Ekman flow that is dynamically coupled to surface wind stress, the ocean energy uptake O in Eq. 1 can be written in terms of a meridional Ekman heat transport F Ek, with O = ·F Ek. The anomalous atmospheric energy balance can be represented as

$∇⋅δ F A +∇⋅δ F R +∇⋅δ F E k =-∇⋅ F S$
(2)

where δ denotes the response to extratropical forcing S. For convenience, δR, with the global mean removed, is expressed as the convergence of a flux such that δR = −·δF R. The AET response (δF A) to the prescribed forcing F S is modulated by the strengths of radiative (δF R) and Ekman (δF Ek) feedbacks.

We compare the tropical responses to the prescribed northward flux in the model with and without Ekman advection to examine the role of meridional Ekman advection on modulating the ITCZ response to the extratropical thermal forcing. In response to the prescribed northward flux F S, the tropical precipitation is clearly shifted to the north in the absence of any ocean dynamics (dashed in Fig. 3d), whereas a much weaker or even reversed response of tropical precipitation is obtained with the Ekman advection (solid in Fig. 3d), consistent with fully coupled model results.

First, consider cases without Ekman advection (i.e., F Ek = 0). As the southern extratropics are cooled, the mid-latitude jet in the SH strengthens and shifts poleward due to enhanced meridional temperature gradients77 (Fig. 4a). The NH presents an opposite response, albeit weaker in magnitude as smaller SST changes are induced in the warmed NH than in the cooled SH (Fig. 3b). This is because the turbulent fluxes require larger SST changes under a colder and drier mean state to balance the prescribed surface heating.22,42 The contrast may be enhanced in comprehensive models since nonlinear cloud radiative effect changes amplify the forcing in the cooling region.41 The strengthened (weakened) eddy fluxes in the SH (NH) extract more (less) energy out of the tropics.78 As the southern (northern) tropics become colder (warmer), the southern (northern) HC becomes stronger (weaker), resulting in an anomalous counter-clockwise HC32,79 (Fig. 4a). This accompanies a northward displaced ITCZ and southward cross-equatorial AET (red in Fig. 5a). Regardless of the amplitude of F S0, the AET response (δF A) in the tropics compensates about 20–25% of the prescribed forcing F S0 while the remainder is compensated by local radiative fluxes (δF R) (Fig. 5a and b). The results are consistent with previous studies.15 As a result, there is a northward shift of the energy flux equator as well as the tropical rainfall, which increases in magnitude as the prescribed forcing is strengthened (dash-dot in Fig. 6).

Next, consider cases with Ekman advection, i.e., F Ek ≠ 0 (refer to Fig. 7). In the tropics, accompanied by a strengthening (weakening) of the southern (northern) HC, the surface easterlies in the SH (NH) become stronger (weaker). The asymmetric zonal surface wind anomalies in the two hemispheres induce an anomalously southward Ekman flow in the tropics that brings about warm (cold) advection in the southern (northern) tropics. This opposes the effect of prescribed extratropical forcing, leading to a reduction in the surface temperature response in the deep tropics as compared to the model without Ekman advection (Fig. 3b). On the other hand, in the extratropics, the surface westerlies are strengthened (weakened) in the SH (NH) associated with a poleward (equatorward) jet shift. An anomalously northward Ekman flow is induced in both hemispheres that brings about cold (warm) advection in the southern (northern) extratropics. This amplifies the effect of the prescribed extratropical forcing, leading to an increase in the surface temperature response in the extratropics (Fig. 3b).

Figure 5c compares the latitudinal profiles of δF A and δF Ek normalized by F S0. It clearly shows that the effects of meridional Ekman heat advection in the tropics and the extratropics counteract each other. The tropical component acts as a negative feedback that partially compensates for the prescribed forcing, whereas the extratropical component acts as a positive feedback that amplifies the prescribed forcing. The extratropical component is larger in the SH than in the NH due to greater jet shifts associated with larger SST changes. Overall, the negative feedback from the tropical component dominates over the positive feedback from the extratropical component, so that the need for the cross-equatorial AET is reduced with Ekman advection. One can define the degree of compensation of the prescribed forcing by the atmospheric and Ekman transport as the total compensation C (i.e., C ≡ (δF A + δF Ek)/F S0) averaged between 5°S and 5°N. The total compensation becomes weaker by less than 10% with the inclusion of Ekman advection, possibly due to the extratropical positive feedback (contrast solid and dash-dot in Fig. 5b). This is in contrast to other fully coupled models where the total compensation is slightly enhanced compared with that in the coupled atmosphere–slab ocean models.61,62,65 It is yet to be examined what determines the total compensation in the coupled system.

In the model with Ekman advection, the total compensation can be partitioned between the contribution from the atmosphere (δF A) and that from the Ekman flow (δF Ek). The fraction of total compensation that is accomplished by the atmospheric transport is the atmospheric compensation C A (i.e., C A ≡ δF A/(δF A + δF Ek)). Similarly, the Ekman compensation C Ek is defined as C Ek ≡ δF Ek/(δF A + δF Ek). By construction, the sum of C A and C Ek is 1. In the model without Ekman advection, C A = 1 and C Ek = 0. For a fixed total compensation C, a larger C Ek indicates that less energy is transported via the anomalous HC, dampening the ITCZ shift. The Ekman compensation reaches up to 70% in the case with the weakest F S0 (Fig. 5d). As a result, the anomalous HC is 50% weaker (Fig. 4), and the energy flux equator is shifted less compared to the model without Ekman advection (Fig. 6a). The tropical precipitation response exhibits little or a slightly southward shift (Figs. 3d and 6b). This is because the Ekman flow induces anomalous warming (cooling) in the southern (northern) deep tropics, leading to an anomalous rising (sinking) motion just south (north) of the equator (Figs. 4b and 7b). Hence, the latitude of the maximum moisture convergence and the energy flux equator decouple. We demonstrate that an inclusion of Ekman advection completely damps or reverses the tropical rainfall response, as in the fully coupled model experiments mentioned previously.53,54,55,56,61,62 Our results suggest that ocean Ekman advection can play a critical role in the coupled climate system.

Complications arise from the competing effects of the tropical and the extratropical components of Ekman-driven heating. As the amplitude of the forcing increases, the positive feedback from the extratropical component is intensified (Fig. 5c), leading to a reduction of Ekman compensation to 30% in the case with the strongest F S0 (Fig. 5d). The results indicate a large dependence of the Ekman advection effect on the response of extratropical jet. The response in the jet latitude is a function of its climatological value,80,81 suggesting a sensitivity of the effect of Ekman advection to the mean state. This may partly explain a large multi-model spread of the partitioning of the energy transport adjustments between atmosphere and ocean.53,54,55,56,61,62 The partitioning is also expected to be sensitive to the location of the prescribed forcing because of large differences among ocean basins in the mean jet latitude, as well as the relative importance between the thermohaline and wind-driven circulation in transporting energy. Moreover, the climatological jet location is sensitive to the strength of transient eddy fluxes,82 which controls the communication between the extratropics and the HC.15 This postulated intimate relationship between δF A and δF Ek should be studied with care in order to better understand the partitioning of energy transport between the atmosphere and ocean.71,83,84

Our simple model that only includes the meridional Ekman advection partly captures the ocean dynamical effect in balancing the external forcing. Schematic that summarizes the effect of Ekman advection is demonstrated in Fig. 8. Future studies need to investigate how the deep return flow and the upwelling effect modify the result by simulating a full dynamical coupling between the Hadley cells and ocean subtropical cells. As the ITCZ migrates away from the equator in response to inter-hemispheric forcing, off-equatorial upwelling of cold waters in the cooled hemisphere intensifies in response to cross-equatorial wind.85 This opposes the negative feedback of tropical Ekman advection. The relative importance of surface Ekman heat advection and the upwelling effect will alter the magnitude of the ITCZ shift. However, if the forcing was large enough that the ITCZ shift accompanies tropical surface westerlies, upwelling will occur near the ITCZ. This increases the ocean energy uptake in the hemisphere with the ITCZ, dampening the ITCZ shift. Another complication may result from the response in subtropical regions of Ekman pumping. That is, a larger (smaller) region of Ekman pumping in the cooled (warmed) hemisphere associated with a poleward expansion (equatorward contraction) of the region of extratropical westerlies may act as a negative feedback, leading to a reduction of the ITCZ shift.

## Concluding remarks

Understanding the dynamics that control tropical rainfall changes is essential for reliable future projections. The zonal-mean atmospheric energy budget has been invoked to understand changes in tropical rainfall, such as its peak location21 or width.86 The energetics framework, in particular, provided a theory for how extratropical inter-hemispheric thermal forcing can displace tropical rainfall, as inferred by paleoclimate records. The framework, however, faces multiple challenges. For a more complete theory, an account of the following factors is necessary: (1) a theory for TGMS, which measures the efficiency with which the HC transports energy,16 (2) a prediction of cloud response based on observed relationships between cloud radiative effect and circulation,44 (3) an expansion to include the zonal dimension to quantitatively predict the regional features, which has recently been attempted by taking into account the zonal component of divergent AET,46 and (4) the effect of ocean dynamics.

Among the aforementioned limitations, we address the issue of partitioning between atmosphere and ocean energy transports. Some recent studies that incorporate fully coupled models indicate that tropical rainfall is not as sensitive to extratropical forcings53,54,55,56,61,62 as suggested by studies that disregard ocean dynamics. We propose that developing a hierarchy of ocean models is a step toward resolving the issue. The largest contributor to the meridional heat transport in the tropical ocean is the Ekman flow that is induced by surface zonal winds. Here, we have considered the meridional Ekman heat advection while neglecting the upwelling effect. (Complexity can be added by accounting for the return flow at depth by incorporating a two-layer ocean model.74) The Ekman advection in the tropics acts as a negative feedback that partially compensates for the prescribed anti-symmetric forcing, whereas the extratropical component acts as a positive feedback that amplifies the prescribed forcing (Fig. 8). In all of the cases under consideration, the negative tropical feedback outweighs the positive extratropical feedback. By partially balancing the prescribed forcing, the meridional Ekman advection reduces the need for AET response and damps the ITCZ shift. This is consistent with a recent study that employs the simplified atmospheric model as in our study but coupled to a full dynamical ocean model.65 The wind-driven ocean circulation is shown to strongly damp ITCZ shifts due to a robust coupling between the atmosphere’s HC and ocean’s subtropical cells through the trade winds. While that study focuses on the negative ocean feedback in the tropics, here we show an additional positive ocean feedback in the extratropics. It remains to be examined how the extratropical component of ocean dynamical feedback responds to the effect of a return flow in the lower layer. The strength of the positive extratropical feedback depends on the response of extratropical jet, which is a function of the mean state.80,81 Hence, the differences in the mean state across models may partly explain the multi-model spread in the degree of ocean circulation compensation.53,54,55,56,61,62 Likewise, the differences in the mean state jet latitude across basins postulates the sensitivity to the location of prescribed forcing. Therefore, a well-coordinated model intercomparison project is needed to better understand how the ocean circulation modulates the tropical response to extratropical forcing.

## Methods

The atmospheric model is a simplified moist GCM where gray radiative transfer is used in which radiative fluxes are only a function of temperature, so that water vapor and cloud radiative feedbacks are eliminated.67,68 Solar radiation is a function of latitude only that mimics the annual mean insolation. There is no seasonal or diurnal cycle in the model. It is coupled to a 50 m aquaplanet slab ocean with no continents or barriers. We note that the results presented herein are nearly independent of the mixed layer depth h. The same experiments with a shallower mixed layer depth of 25 m are performed whose results are compared in Figs. 5 and 6. This insensitivity is because the heating induced by the Ekman flow is independent of h due to the proportionality of v Ek to h. However, the results may be modified with a temporally varying forcing or an inclusion of seasonal cycle depending on how the forcing is rectified.87

The surface zonal wind stress τ x is computed via a bulk formula as $ρ a C D w 10 u 10$, with the density of air ρ a=1.2 kg m−3, a bulk transfer coefficient for momentum C D = 1.3 × 10−3, and the scalar horizontal wind speed $w 10$ and the zonal wind velocity u 10 at a height of 10 m.

All experiments are integrated for 20 years, with a spin-up period of 10 years.

### Data availability

The data that support the findings of this study are available from the corresponding author upon request.

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## Acknowledgements

The authors gratefully acknowledge three anonymous reviewers whose insightful comments led to significant improvements to the manuscript. S.M.K. and Y.S. are supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (2016R1A1A3A04005520). S.P.X. is supported by the US National Science Foundation (1637450).

## Author information

### Affiliations

1. #### School of Urban and Environmental Engineering, Ulsan National Institute of Science and Technology, UNIST-gil 50, Ulsan, 44919, Republic of Korea

• Sarah M. Kang
•  & Yechul Shin
2. #### Scripps Institution of Oceanography, University of California San Diego, 9500 Gilman Drive, MC 206, La Jolla, CA, 92093, USA

• Shang-Ping Xie

### Contributions

S.M.K. and S.P.X. designed the project. Model experiments are performed by Y.S. and data were analyzed by all authors. All authors contributed during the writing of the paper.

### Competing interests

The authors declare no competing financial interests.

### Corresponding author

Correspondence to Sarah M. Kang.


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## Independent sets into which all the vertices of the graph can be split

How to prove that if $$G$$ is an acyclic transitive digraph, then the least independent sets into which all vertices of G can be divided is equal to the size of the longest paths to $$G$$?

## Semi streaming algorithm for 2 vertices connectivity

Let $$G=(V,E)$$ be an undirected graph. given a pair of vertices $$s,t \in V$$, how can we construct a semi-streaming algorithm which determines is $$s$$ and $$v$$ are connected? Is there any way to construct such an algorithm which scans the input stream only once?

Note that a semi-streaming algorithm is presented an arbitrary order of the edges of $$G$$ as a stream, and the algorithm can only access this input sequentially in the order it is given; it might process the input stream several times. The algorithm has a working memory consisting of $$O(n⋅polylogn)$$.

## Remove vertices

Given an undirected graph G(V,E), find the minimum number of vertices to remove so that the Edge Set of the graph becomes $$\phi$$.

In other words: If we remove one vertex we remove that vertex along with all edges connected to that vertex. Find the minimum number of vertices to remove so that there are no edges left in G after their removal.

## Finding maximum subgraph with vertices of degree at most k

Let $$G = (V, E)$$ be an undirected graph and $$U \subseteq V$$ some subset of its vertices. An induced graph $$G[U]$$ is graph created from $$G$$ by removing all vertices that are not part of the set $$U$$.

I want to find a polynomial time algorithm that has graph $$G = (V, E)$$ and integer $$k$$ as input and returns a maximum set $$U \subseteq V$$ with largest size such that all vertices of $$G[U]$$ have degree at most $$k$$.

My idea with greedy algorithm that removes vertices with largest degree or vertices connected with most vertices with degree greater than $$k$$ doesn’t work.

Does anyone know how to solve this problem in polynomial time?

## Paint graph ordered vertices

I have the following question: Given graph $$G=(V,E)$$ with given order on its vertices, I mean $$v_1 I need to find minimal colors ordered paining of the graph vertices s.t neighbor vertices don’t have the same color, I mean: $$(v_1,v_2,…,v_j)$$ – painted in fist color and non of them are neighbors. $$(v_{j+1},…,v_l)$$ – painted in second color and non of them are neighbors. and so on.

The complexity needs to be $$O(V+E)$$. Anyone have an idea?

## Navmesh awkward path generation with string pulling due to “inner” vertices

I’ve identified a problem and a possible solution related to navmesh-based pathfinding. Before diving in, I’ll preface my post with some questions to keep in mind as you read:

• Is this a known problem that people have solved before?
• Is there a term for the problem that could help me search for information related to it?
• Is the solution I came up with an existing idea? If so is there a name for the algorithm or some other search term I could use to find more information?
• Is there a better solution? If so, please point me to it.

For reference, I’m using images from http://jceipek.com/Olin-Coding-Tutorials/pathing.html#navigation-meshes and generally following the advice laid out there.

tl;dr of that blog post is

Decompose your walkable area into a navmesh, treating convex polygons as nodes and their borders as edges so that you can perform an A* search to get from point A to point B. To translate from “node ids” back to real points, use string-pulling.

Here’s a copy of the example space:

And an example generated path after performing string pulling:

So far so good.

But I realized this approach generates an awkward path in a situation like this:

In this situation, a trio of nodes are all adjacent to each other, and so the A* will generally choose a path directly from the starting node to the ending node, despite an intuitive understanding that the agent can move in a straight line from A to B which travels through a different polygon.

I’ve been working on a solution to this problem and so far my best idea is to apply a transformation to the nav mesh. My description of this will be a little hazy as I’m making up terminology to describe the approach…

• Define a shared edge as a line segment that is shared by two convex polygons in the navmesh. Maybe a.k.a. a “portal” for string-pulling purposes.
• Define an inner vertex as a vertex in the navmesh for which all attached line segments are “shared edges”. The vertex in the center of the three polygons in the image above is an inner vertex.
• Identify an inner vertex. Follow its attached shared edges to what I’ll call neighbor vertex. (possible improvement; If the neighbor vertex is also an inner vertex, recurse to its neighbors until all of the neighbors are non-inner.)
• Remove all shared edges from the navmesh that were traversed in the previous step, forming a new polygon whose border is defined by the neighbor vertices in the previous step. Redefine the edges accordingly (I’ll hand-wave that)
• Repeat until no inner vertices remain.

The result of this on the example above should result in this:

And with the same A-B path from before, the string-pulling should now result in a straight line:

I believe that as long as the navmesh has no inner vertices, all paths generated with the approach described in the linked blog post should seem “natural” and not have any surprise corners in what seems like open space.

Per my questions at the beginning of this post, I’m looking for more information, e.g. has anybody else solved this problem before, is there a better way to do it, and is there even a name/term for this problem?

## Given a tournament with $2^n$ vertices, show that there is a sub-tournament with at least $n + 1$ vertices that is acyclic

So a tournament is just a complete directed graph, I believe.

I’m having trouble proving this problem. I know it is induction however. I was thinking the base case is $$2^1$$ vertices, and therefore we have a sub-tournament of 1 + 1 vertices, which holds.

Then in our induction step we have to show $$2^{n + 1}$$. But I’m not sure how to approach this. Any ideas would be extremely appreciated.

## find shortest paths from source to all vertices using Dijkstra’s Algorithm?

For Dijkstra’s,i can find shortest paths from source to all vertices in the given graph but how can i calling the algorithm |V| times taking each vertex as a source and store all tables ??? For example : What is the shortest path from 1 to 4? You need to print the value and the exact path vertices starting from 1 and ending at 4.

## write a c++ code to solve shortest path between any two input vertices

Implement Floyd’s Algorithm and Dijkstra’s algorithm using all vertices as sources to solve All- pairs shortest path problem. For Dijkstra’s, you need to call the algorithm |V| times taking each

vertex as a source and store all tables. As a hint you may define T, presented in class, as a two- dimensional array, instead of being a one-dimensional array, of structures. You will read a graph

with random edge weights from a file and store the graph information in an adjacency matrix. Then, you need to answer any query asking for the shortest path between any two input vertices. You need to print the path from Floyd’s and Dijkstra’s algorithms. Your program should ask the user for the name of the input file.

## Check if a pair of vertices belongs to the min-cut

Given a digraph $$G = (V,A)$$ with a source $$s \in V$$ and a sink $$t \in V$$, I need to adapt the graph to known if a pair of vertices $$u \in V$$ and $$v \in V$$ belongs to the min-cut $$S$$ between $$s$$ and $$t$$. That is, a new vertex $$a$$ belongs to the min-cut $$S$$ if and only if $$u \in S$$ and $$v \in S$$. This new vertex must preserve the original max-flow from $$s$$ to $$t$$. I tried creating new arcs $$(u,a)$$ and $$(a,v)$$ with infinity capacities, but while $$a$$ belongs to $$S$$ when $$u \in S$$, this immediately forces $$v$$ to belong to $$S$$ also, which is not necessarily true if the max-flow was computed in the original graph $$G$$. So, is there a way to force a new node to be in the min-cut set $$S$$ if a pair of vertices belongs to $$S$$?

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So I’ve had a little downtime over the last few days in the office and was thinking about my cost of earning an MBA. This mini-analysis ended up taking on a life of its own and it turned into a pretty sophisticated look at figuring out what am I giving up and how long (and how much) it is going to take to pay off. I realize that $is not the (only) reason that people go to b-school, but it's definitely an important factor. As a very analytical person, I thought it was important to dig deep on this topic. I have put together what I think is a comprehensive MBA return on investment (ROI) tool/model (more specifically, I used net present value, NPV). It looks at the MBA and non-MBA career paths and identifies not only when you’ll “breakeven” but also how your compensation will look over an extended period of time. I believe there are some other tools available online, but I don’t think that any of them offer much flexibility and are way to simplistic for a very important and complicated decision. Since I’ve gained so much from gmatclub, I wanted to share this tool so that my fellow g-clubbers can tailor it based on personalized life plans. For example, some people want to go into i-banking. This particular job function has a pretty unique compensation path. Moreover, most people that (want to) go into banking, only do so for a few years before jumping to investment management or private equity or some other excursion. Therefore, it wouldn’t make sense to say, “My compensation is going to be$350k for the next 12 years.” Other folks want to go into marketing, consulting, operations, strategy, etc. - areas that have different compensation tracks than investment banking. This tool is designed for any person starting with/transitioning to any career path.

I realize this may not be perfect, but I think it will be very informative regardless of your interests. This should help you whether you’re considering an elite vs. a trans-elite or a tier-two school vs. a regional program. It will also be helpful for those that are deciding between a part-time or full-time program. This will also help high-income earners that are deciding whether to go back to school at all. Ultimately, I wanted to create something that is user friendly and allows for almost an unlimited number of scenarios. If you have any questions, feel free to ask. Enjoy and let me know what you think.

Cheers,
ryguy

EDIT - Version 2.0 Available Now!

I made a handful of changes since I created this model last summer, but below are the two major changes that I have made:

1) Updated fields to incorporate internship earnings

2) Ability to easily compare a part-time v. full-time MBA program (from a financial perspective)

A couple of comments and feedback that were suggested from last time that I did not specifically address (and my reasons for choosing not to do so).

1) Debt service of student loans. So here’s the deal. This model assumes that tuition is paid upfront (i.e., during school). This is unrealistic for the vast majority of folks on the board here, and probably most students in general. However, I prefer to take a highly conservative approach when it comes to figuring out when I will break even. I suppose this is a worst-case scenario (i.e., makes the payback period look longer than it should), but in my mind, I’d rather err on the side of a longer payback period than a shorter payback period.

So why does this matter when tuition is paid? It is more expensive to pay for tuition with today’s dollars than it is X years down the road. For instance, if you are on a 10-yr repayment plan paying $600 a month -$600 in today’s terms is a lot more money than it is 5, 6, or 7 years from now (and for simplicity’s sake, let’s ignore fluctuations in the value of the US dollar). If you don’t believe me, ask yourself this: Would you rather I gave you $600 today or$600 in 6 years from now. Good, I’m glad we’re on the same page. For more detail on this, I explain this in general assumption #3 on the Instructions worksheet. In that section, I’ve also included instructions for a scenario that could be applicable for folks that go into investment banking, private equity, or any other very high paying post-MBA job.

2) Taxes. I received some comments saying that tuition is paid with after-tax dollars, but salary is listed in before tax dollars. Sure, fair enough. If you think that it is appropriate to adjust for this, multiple your salary by (1 minus your expected tax rate) and plug that in for your post-grad salary. I chose not to do this for the base case scenario, but again, I’ve tried to make this model flexible enough where you can put in whatever assumptions YOU think are appropriate.

Updated directions are included on the first worksheet of the excel file. Let me know if you have any questions. I’ll be happy to address any issues/comments that may be related to your particular circumstances. Also, if you think there are any improvements that could be made to this for future versions, I’ll see what I can do to incorporate them. And again, I'm human, so if you find any errors, let me know and I'll correct them. Enjoy and let me know what you think.
Attachments

File comment: Value of an MBA v2.0
Value of an MBA v2.0.xls [70 KiB]

File comment: Value of an MBA
Value of an MBA.xls [62 KiB]

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The True Value of YOUR MBA: 103-t64239
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Re: The True Value of YOUR MBA [#permalink]  27 May 2008, 07:23
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Re: The True Value of YOUR MBA [#permalink]  27 May 2008, 07:30
Good spreadsheet, though it makes me feel sad. My breakeven point is a long way off
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Re: The True Value of YOUR MBA [#permalink]  27 May 2008, 08:15
great model. i had set up a similar one before applying but yours is even more detailed.

the only thing i would add is that these types of models have a difficult time incorporating option value. there's obvious option value in completing an MBA program and getting a job you otherwise would not have been able to get, but there is a lot of "unknown" option value as well. to the extent that you meet the right people and get connected into amazing opportunities, so much the better. my point is that if the model is close on an after cost basis (as might be the case if you are not an aspiring finance baller), it's probably worth rolling the dice if you can get into a good school -- you never know what might come your way.
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Re: The True Value of YOUR MBA [#permalink]  17 Jul 2008, 19:14
thanks for providing this.
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Re: The True Value of YOUR MBA [#permalink]  18 Jul 2008, 07:11
Ryguy,

Great tool, very comprehensive!

Thanks!

~Sam
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Re: The True Value of YOUR MBA [#permalink]  18 Jul 2008, 12:58
great tool! How you would suggest us PT'ers edit it to make our situation reflect the results!

Either way, +1!!
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Re: The True Value of YOUR MBA [#permalink]  18 Jul 2008, 15:30
2
KUDOS
zakk wrote:
great tool! How you would suggest us PT'ers edit it to make our situation reflect the results!

Either way, +1!!

Great question, perhaps something that I can work on for Version 2.0.

I think the easiest way would be to manually override the cash flows for the first three years. For instance, the average full-time MBA does not have any income in Yr 1 and Yr 2 (with the exception of an internship), so his/her cash flow is negative and simply the amount of tuition. In your case, your tuition will be offset by your income (sweet!), so you will have positive cash flow in those years (salary - tuition + any company reimbursement = cash flow). You can use that as a starting point for your years 1 and 2. The other adjustment that you will need to make is for "salary upon graduation." I assume that the PT program is three years, instead of two, so you will need an override adjustment in year 3 (again, this is the aforementioned formula). From there it should work, since the stages are set up to increase your salary by the percentage that you designate.

Hope that helps. Let me know if you have any more questions or would like some clarity on this.
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Re: The True Value of YOUR MBA [#permalink]  20 Jul 2008, 15:44
great tool! +1!
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Re: The True Value of YOUR MBA [#permalink]  20 Jul 2008, 17:34
yikes, 6-7 years to recoup my investment in the best case scenario. Anybody want to try making an argument about how pursuing an MBA has significant intangible value not reflected in ryguy's mad awesome spreadsheet?

Certainly puts a lot of things into perspective.
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Re: The True Value of YOUR MBA [#permalink]  21 Jul 2008, 15:44
solaris1 wrote:
yikes, 6-7 years to recoup my investment in the best case scenario. Anybody want to try making an argument about how pursuing an MBA has significant intangible value not reflected in ryguy's mad awesome spreadsheet?

Certainly puts a lot of things into perspective.

First off, hopefully you are also looking to enter a field that in some way you will find more enjoyable post-mba. That is some of the intangible value. Then, on the financial side you need to consider the spreadsheet is the average number. You will do better or worse than this, but will almost certainly not hit the exact number. The big thing here is,as "Sudden" said above:
"the only thing i would add is that these types of models have a difficult time incorporating option value"
I found this very well stated - Option value is very tough to predict. Basically, while not necessarily likely, you have a vastly increased chance of doing something that will make you filthy rich.

I think that sums up the 2 main intangibles - did I miss any?
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Re: The True Value of YOUR MBA [#permalink]  23 Jul 2008, 11:46
Great spreadsheet ryguy! Haven't had a chance to really pour into it but looks like a fun tool.

I would like to add a potential question to those that are trying to calculate a precise ROI using this model. How does tax-affecting cash flows change the end result of your calculations? I'm not sure if the impact will be material or not but I'm looking forward to playing around with the model a bit to see.
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Re: The True Value of YOUR MBA [#permalink]  24 Jul 2008, 15:34
ryguy904 wrote:
zakk wrote:
great tool! How you would suggest us PT'ers edit it to make our situation reflect the results!

Either way, +1!!

Great question, perhaps something that I can work on for Version 2.0.

I think the easiest way would be to manually override the cash flows for the first three years. For instance, the average full-time MBA does not have any income in Yr 1 and Yr 2 (with the exception of an internship), so his/her cash flow is negative and simply the amount of tuition. In your case, your tuition will be offset by your income (sweet!), so you will have positive cash flow in those years (salary - tuition + any company reimbursement = cash flow). You can use that as a starting point for your years 1 and 2. The other adjustment that you will need to make is for "salary upon graduation." I assume that the PT program is three years, instead of two, so you will need an override adjustment in year 3 (again, this is the aforementioned formula). From there it should work, since the stages are set up to increase your salary by the percentage that you designate.

Hope that helps. Let me know if you have any more questions or would like some clarity on this.

Worked like a charm! Just took a bit of work.

Looks like 8 years to work mine off, but the opportunity this presents is simply too much to pass over, even if it is a year later than I'd like to start.

Thanks again!
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Re: The True Value of YOUR MBA [#permalink]  24 Jul 2008, 17:53
zakk wrote:
Worked like a charm! Just took a bit of work.

Looks like 8 years to work mine off, but the opportunity this presents is simply too much to pass over, even if it is a year later than I'd like to start.

Thanks again!

Hey Zakk,

Glad that worked. I will definitely incorporate a PT feature in the next version.

As far as *your* payback goes, there is NOTHING wrong with 8 years. The way that NPV and the time value of money works is that if you are comparing project A (or no MBA in this case)and project B (MBA), you will choose the project that has the highest NPV (which for you is earning an MBA). If it said the payback was 45 years and you are 70 years old now, well, then that would be a problem. But for any of us in our 20's or even early 30's, there is nothing wrong with an 8 year or even a 10 year payback. If you include the intangibles, which unfortunately I'm not capable of quantifying - your payback will be even sooner!

All the best,
ryguy
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Re: The True Value of YOUR MBA [#permalink]  30 Jul 2008, 14:00
+1 for you, RyGuy. Very, very helpful stuff. Must have taken quite a bit of time and effort to set this up, so much appreciated!
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Re: The True Value of YOUR MBA [#permalink]  02 Aug 2008, 11:21
Nice spreadsheet!! Just one thing to note, your salaries are pre-tax and the tuition cost are post-tax. It may make more sense to convert tuitions cost to a pre-tax number (e.g $45K becomes 45/(1-0.40) = 75) Crap, my breakeven is 10 years + . Manager Joined: 06 Jan 2008 Posts: 215 Followers: 1 Kudos [?]: 53 [0], given: 1 Re: The True Value of YOUR MBA [#permalink] 03 Aug 2008, 14:14 Nice job. I did a similar exercise for my wife's medical school and physician career. Considers yourselves lucky, ladies and gentlemen, in an unspecialized medical practice (ie internal medicine), one never breaks even. Also, I did not notice a section in this model for debt service, which would add at least a few years onto the breakeven. Director Joined: 25 Dec 2007 Posts: 524 Schools: Harvard '11 Followers: 9 Kudos [?]: 63 [0], given: 28 Re: The True Value of YOUR MBA [#permalink] 03 Aug 2008, 16:19 In a 40+ year career, a breakeven point of 10 years is not bad at all! Let's say you work for 3 years, get your MBA, and then spend the next 10 years getting back to even. Then you spend the next 25 years making great money and living the good life. Intern Joined: 26 Jan 2008 Posts: 19 Followers: 1 Kudos [?]: 9 [2] , given: 0 Re: The True Value of YOUR MBA [#permalink] 10 Aug 2008, 06:20 2 This post received KUDOS interesting analysis, thank you. but why you are so pessimistic about your future? growth rates look a quite low. is it really than even in 2020 (12 years ahead) you can get only 186k with MBA and 90 without? what is your country and industry? I think you should believe in yourself more! P.S. what about taxation and inflation? do you estimate incomes after it? What about economic cycle? maybe changing career in the future? P.S.2: My ROI is definitely very negative, but I'm not worried about it. I think education can not be measured just in money. personal effect more important and can not be measured. Manager Joined: 11 Apr 2008 Posts: 202 Followers: 2 Kudos [?]: 17 [0], given: 1 Re: The True Value of YOUR MBA [#permalink] 12 Aug 2008, 10:45 This is a very good source of information but I think it is highly empirical, So I have several questions and I hope somebody will address to these. 1. Does this equation holds good for all MBA program? Or only from top 15 schools from US? 2. And the break even has been calculated taking 60k as current salary. What about those cases or people who are already making more than 60K? 3. Does this calculation addresses the increased tax and expenditure due to increase in salary in future? sorry to ask such basic or preliminary questions as I am from pure science background ( a very little idea on business and finance) thanks all _________________ Nobody dies a virgin, life screws us all. Current Student Joined: 27 Jul 2007 Posts: 872 Location: Sunny So Cal Schools: CBS, Cornell, Duke, Ross, Darden Followers: 12 Kudos [?]: 193 [2] , given: 4 Re: The True Value of YOUR MBA [#permalink] 12 Aug 2008, 11:21 2 This post received KUDOS subarao wrote: This is a very good source of information but I think it is highly empirical, So I have several questions and I hope somebody will address to these. 1. Does this equation holds good for all MBA program? Or only from top 15 schools from US? 2. And the break even has been calculated taking 60k as current salary. What about those cases or people who are already making more than 60K? 3. Does this calculation addresses the increased tax and expenditure due to increase in salary in future? sorry to ask such basic or preliminary questions as I am from pure science background ( a very little idea on business and finance) thanks all The numbers that I have plugged in are as a sample (i.e., illustrative purposes only) so that you know which data points you should enter ON YOUR OWN. As I said in my original post and in the directions tab, this is supposed to look at the value of YOUR MBA. So plug in YOUR numbers. It doesn't matter if you go to top 50 or top 5, it's all based on the assumptions that you put in. However, if you are assuming$140,000 starting salary for a top 50 school, well then "garbage in, garbage out."

Currently it does not account for taxes. The easy way to get around that is to simply enter all of your info in net tax (after tax) dollars.

And again, this is not designed to be the ultimate solution for everyone. The main point is to give you an idea what your situation looks like. If your NPV is 5 years - great. If it's 10, that's also great, since we are all concerned with the long run. Even if it's >20years, that may be OK for some people because of the intangible aspects of going to school (e.g., network, formal business education, career switch, etc.).

Hope that helps.
Re: The True Value of YOUR MBA   [#permalink] 12 Aug 2008, 11:21

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text:
# nLab topological cofiber sequence

Contents

under construction

# Contents

## Idea

We discuss in detail the realization of the abstract concept of cofiber sequences in its explicit incarnation in point-set topology, the way it is traditionally presented in topology textbooks.

Hence we use the concepts of homotopy equivalence instead of weak homotopy equivalence. For discussion using the latter in the context of the classical model structure on topological spaces see instead at Introduction to Homotopy Theory the section Homotopy fiber sequences.

## Background

Throughout, write $I \coloneqq [0,1] \subset \mathbb{R}$ for the closed interval equipped with its Euclidean metric topology.

### Homotopy equivalences

###### Definition

For $f,g\colon X \longrightarrow Y$ two continuous functions between topological spaces $X,Y$, then a homotopy

$\eta \colon f \,\Rightarrow_L\, g$
$\eta \;\colon\; X \times I \longrightarrow Y$

out of the standard cylinder object over $X$: the product space of $X$ with the Euclidean closed interval, such that this fits into a commuting diagram of the form

$\array{ X \\ {}^{\mathllap{(id,\delta_0)}}\downarrow & \searrow^{\mathrlap{f}} \\ X \times I &\stackrel{\eta}{\longrightarrow}& Y \\ {}^{\mathllap{(id,\delta_1)}}\uparrow & \nearrow_{\mathrlap{g}} \\ X } \,.$

(graphics grabbed from J. Tauber here)

###### Definition

(homotopy equivalence)

A continuous function $f \colon X \to Y$ is called a homotopy equivalence if there exist

1. a continuous function $g \colon Y \to X$;

2. homotopies (def. ) of the form

$g \circ f \Rightarrow id_X \phantom{AAAA} f \circ g \Rightarrow id_Y$

###### Example

(contractible topological space)

A topological space $X$ is called contractible if the unique map to the point space $X \to \ast$ is a homotopy equivalence (def. ).

### Cofibrations

###### Definition

(Hurewicz cofibration)

A continuous function $i \colon A \longrightarrow X$ is called a Hurewicz cofibration or just cofibration if it satisfies the homotopy extension property in that:

• for any topological space $Y$,

• for all continuous functions $f \colon A\to Y$, $\tilde{f}:X\to Y$ such that $\tilde{f}\circ i=f$

$\array{ A &\stackrel{f}{\to}& Y \\ {}^{\mathllap{i}}\downarrow & \nearrow_{\mathrlap{\tilde f}} \\ X }$
• and any left homotopy $F \colon A\times I\to Y$ such that $F(-,0)=f$

there is a homotopy $\tilde{F} \colon X\times I\to Y$ such that

• $\tilde{F}\circ(i\times id_I)=F$

$\array{ A \times I &\overset{F}{\longrightarrow}& Y \\ {}^{\mathllap{i \times id_I}}\downarrow & \nearrow_{\mathrlap{\tilde F}} \\ X \times I }$
• and $\tilde{F}(-,0)=\tilde{f}$

$\array{ A &\overset{id_A \times const_0}{\longrightarrow}& A \times I &\overset{F}{\longrightarrow}& Y \\ {}^{\mathllap{i \times id_I}}\downarrow && {}^{\mathllap{i \times id_I}}\downarrow & \nearrow_{\mathrlap{\tilde F}} \\ X &\underset{id_X \times const_0}{\longrightarrow}& X \times I } \phantom{AAA} = \phantom{AAA} \array{ \array{ A &\stackrel{f}{\to}& Y \\ {}^{\mathllap{i}}\downarrow & \nearrow_{\mathrlap{\tilde f}} \\ X }cco }$

If the image $i(A) \subset X$ is a closed subset, then $i$ is called a closed Hurewicz cofibration.

###### Example

(retracts of relative cell complex inclusions are closed Hurewicz cofibrations)

If $A$ is a topological space and $i \colon A \hookrightarrow X$ is a relative cell complex inclusion, then $i$ is a closed Hurewicz cofibration (def. ).

Also every retract of such $i$ (this sense) is a closed Hurewicz cofibration.

### Pushouts / space attachments

Cofiber sequences (below) are constructed by iterated pushouts in the category Top of topological spaces with continuous functions between them (space attachments), see at Top – Universal constructions.

###### Example

(quotient space by a subspace)

Let $X$ be a topological space and $A \subset X$ a non-empty subset. Consider the equivalence relation on $X$ which identifies all points in $A$ with each other. The resulting quotient space is denoted $X/A$.

Notice that $X/A$ is canonically a pointed topological space, with base point the equivalence class $A/A \subset X/A$ of $A$.

If $A = \emptyset$ is the empty space, then one defines

$X/\emptyset \coloneqq X_+ \coloneqq X \sqcup \ast$

to be the disjoint union space of $X$ with the point space. This is no longer a quotient space, but both constructions are unified by the pushout $i \colon A \to X$ along the map $A \to \ast$, equivalently the cokernel of the inclusion:

$\array{ A &\overset{i}{\hookrightarrow}& X \\ \downarrow &(po)& \downarrow \\ \ast &\longrightarrow& X/A } \,.$
###### Example

(union of two open or two closed subspaces is pushout)

Let $X$ be a topological space and let $A,B \subset X$ be subspaces such that

1. $A,B \subset X$ are both open subsets or are both closed subsets;

2. they constitute a cover: $X = A \cup B$

Write $i_A \colon A \to X$ and $i_B \colon B \to X$ for the corresponding inclusion continuous functions.

Then the commuting square

$\array{ A \cap B &\longrightarrow& A \\ \downarrow &(po)& \downarrow^{\mathrlap{i_A}} \\ B &\underset{i_B}{\longrightarrow}& X }$

is a pushout square in Top (see there).

By the universal property of the pushout this means in particular that for $Y$ any topological space then a function of underlying sets

$f \;\colon\; X \longrightarrow Y$

is a continuous function as soon as its two restrictions

$f\vert_A \;\colon\; A \longrightarrow Y \phantom{AAAA} f\vert_A \;\colon\; B \longrightarrow Y$

are continuous.

###### Proof

Clearly the underlying diagram of underlying sets is a pushout in Set. Therefore (by this prop.) we need to show that the topology on $X$ is the final topology induced by the set of functions $\{i_A, i_B\}$, hence that a subset $S \subset X$ is an open subset precisely if the pre-images (restrictions)

$i_A^{-1}(S) = S \cap A \phantom{AAA} \text{and} \phantom{AAA} i_B^{-1}(S) = S \cap B$

are open subsets of $A$ and $B$, respectively.

Now by definition of the subspace topology, if $S \subset X$ is open, then the intersections $A \cap S \subset A$ and $B \cap S \subset B$ are open in these subspaces.

Conversely, assume that $A \cap S \subset A$ and $B \cap S \subset B$ are open. We need to show that then $S \subset X$ is open.

Consider now first the case that $A;B \subset X$ are both open open. Then by the nature of the subspace topology, that $A \cap S$ is open in $A$ means that there is an open subset $S_A \subset X$ such that $A \cap S = A \cap S_A$. Since the intersection of two open subsets is open, this implies that $A \cap S_A$ and hence $A \cap S$ is open. Similarly $B \cap S$. Therefore

\begin{aligned} S & = S \cap X \\ & = S \cap (A \cup B) \\ & = (S \cap A) \cup (S \cap B) \end{aligned}

is the union of two open subsets and therefore open.

Now consider the case that $A,B \subset X$ are both closed subsets.

Again by the nature of the subspace topology, that $A \cap S \subset A$ and $B \cap S \subset B$ are open means that there exist open subsets $S_A, S_B \subset X$ such that $A \cap S = A \cap S_A$ and $B \cap S = B \cap S_B$. Since $A,B \subset X$ are closed by assumption, this means that $A \setminus S, B \setminus S \subset X$ are still closed, hence that $X \setminus (A \setminus S), X \setminus (B \setminus S) \subset X$ are open.

Now observe that (by de Morgan duality)

\begin{aligned} S & = X \setminus (X \setminus S) \\ & = X \setminus ( (A \cup B) \setminus S ) \\ & = X \setminus ( (A \setminus S) \cup (B \setminus S) ) \\ & = (X \setminus (A \setminus S)) \cap (X \setminus (B \setminus S)) \,. \end{aligned}

This exhibits $S$ as the intersection of two open subsets, hence as open.

A general abstract fact about pushouts which we will use repeatedly in the proofs below is the following:

###### Proposition

(pasting law)

Consider a diagram in Top (or in any other category) of the following form:

$\array{ &\longrightarrow& &\longrightarrow& \\ \downarrow &(po)& \downarrow && \downarrow \\ &\longrightarrow& &\longrightarrow& } \,,$

where the left square is a pushout.

Then: The total rectangle is a pushout precisely if the right square is.

We need the following two facts (prop. prop. ) regarding the stability of cofibrations under pushout.

###### Proposition

(pushout of cofibrations)

Let $A$ be a topological space and let $A \hookrightarrow X$ be a closed Hurewicz cofibration.

Then for every continuous function $f \colon A \to Y$, the pushout $f_\ast i$ in

$\array{ A &\overset{f}{\longrightarrow}& Y \\ {}^{\mathllap{i}}\downarrow &(po)& \downarrow^{\mathrlap{f_\ast i}} \\ X &\longrightarrow& X \underset{A}{\sqcup} Y }$

is also a closed Hurewicz cofibration. Moreover, if in addition

1. $i$ is a retract of a relative cell complex inclusion, then so is $f_\ast i$,

2. $i$ is a homotopy equivalence, then so is $f_\ast i$.

Prop. is a consequence of the existence of the Strøm model structure $Top_{Strom}$ and the classical model structure on topological spaces $Top_{Quillen}$ and of the fact that the identity functors $Top_{Strom} \underoverset{\underset{id}{\longrightarrow}}{\overset{id}{\longleftarrow}}{\bot} Top_{Quillen}$ constitute a Quillen adjunction.

###### Proposition

(quotient by contractible closed subspace)

Let $i \colon A \longrightarrow X$ be a closed Hurewicz cofibration.

If $f \colon A \to Y$ is a homotopy equivalence (def. ) then its pushout $i_\ast f$ in

$\array{ A &\overset{i}{\longrightarrow}& X \\ {}^{\mathllap{f}}\downarrow &(po)& \downarrow^{\mathrlap{i_\ast(f)}} \\ Y &\longrightarrow& Y \underset{A}{\sqcup} X }$

is also a homotopy equivalence.

In particular if $A$ is a contractible topological space (example ), then the coprojection

$X \longrightarrow X/A$

to the quotient space (example ) is a homotopy equivalence.

Prop. is again a consequence of the existence of the Strøm model structure $Top_{Strom}$: It is the statement that $Top_{Strom}$ is a left proper model category which follows (this cor.) since all its objects are evidently cofibrant. An elementary proof in point-set topology is offered in (Hatcher, prop. 0.17)

## Mapping cones

###### Definition

(topological cylinder and cone)

Let $X$ be a topological space. Then

1. the standard cylinder on $X$ is the product topological space

$Cyl(X) \coloneqq X \times [0,1]$
2. the standard cone on $X$ is the quotient space (example )

$Cone(X) \coloneqq Cyl(X) / (X \times \{0\})$

of the standard cylinder by the subspace $X \times \{0\} \subset X \times [0,1]$.

Equivalently this is the following pushout in Top

$\array{ X \times \{0\} &\hookrightarrow& Cyl(X) \\ \downarrow &(po)& \downarrow \\ \ast &\longrightarrow& Cone(X) }$
###### Lemma

Let $X$ be a topological space. Then the canonical inclusions

$X \longrightarrow Cyl(X) \phantom{AAAA} \ast \longrightarrow Cone(X)$

are homotopy equivalences. Hence every cone is a [contractible topological space]] (example ).

###### Definition

(mapping cylinder and mapping cone)

Let $f \colon X \to Y$ be a continuous function between topological spaces. Then

1. the mapping cylinder of $f$ is the space attachment

$Cyl(f) \coloneqq Y \cup_f Cyl(X)$

of $Y$ with the cylinder on $X$, according to def. , hence the following pushout in Top

$\array{ X &\overset{f}{\longrightarrow}& Y \\ \downarrow &(po)& \downarrow \\ Cyl(X) &\longrightarrow& Cyl(f) }$
2. the mapping cone of $f$ is the space attachment

$Cone(f) \coloneqq Y \cup_f Cone(X)$

hence the following pushout in Top:

$\array{ X &\overset{f}{\longrightarrow}& Y \\ \downarrow &(po)& \downarrow \\ Cone(X) &\longrightarrow& Cone(f) }$
###### Remark

In summary, def. and def. say that for $f \colon X \to Y$ a continuous function then we have a pasting of pushout diagrams in Top of the following form:

$\array{ && X &\stackrel{f}{\to}& Y \\ && \downarrow^{\mathrlap{i_1}} &(po)& \downarrow \\ X &\stackrel{i_0}{\to}& Cyl(X) &\to & Cyl(f) \\ \downarrow &(po)& \downarrow &(po)& \downarrow \\ {*} &\to& Cone(X) &\to& Cone(f) } \,.$

Since $X \to Cone(X)$ is a closed Hurewicz cofibration , the pasting law together with prop. therefore implies that also $Y \to Cone(f)$ is a closed Hurwicz cofibration.

###### Example

(suspension)

For $X$ a topological space, then the mapping cone (def. ) of the unique function $X \to \ast$ to the point space is

$S X \coloneqq Cone(X \to \ast) \simeq Cone(X) \underset{X}{\sqcup} Cone(X) \,.$

This is called the suspension of $X$.

The mapping cone of a map $X \to Y$ is to be thought of as the homotopy-quotient of the $Y$ by $X$, as opposed to be the naive quotient. This is made precise by the following two statements, lemma and lemma :

###### Lemma

If $f \colon X \to Y$ is a closed Hurewicz cofibration (def. ), then the coprojection

$Cone(f) \longrightarrow Y/f(X)$

from its mapping cone (def. ) to the naive quotient space (example ) is a homotopy equivalence.

###### Proof

Consider the diagram

$\array{ X &\overset{f}{\longrightarrow}& Y \\ \downarrow &(po)& \downarrow \\ Cone(X) &\longrightarrow& Cone(f) \\ \downarrow &(po)& \downarrow \\ \ast &\longrightarrow& Cone(f)/Cone(X) }$

Since $f$ is a closed Hurewicz cofibration, so is $Cone(X) \to Cone(f)$, by prop. . Since $Cone(X) \to \ast$ is a homotopy equivalence (lemma ), so is $Cone(f) \to Cone(f)/Cone(X)$, by prop. . But by lemma there is also a homeomorphism $Cone(f)/Cone(X) \simeq Y/f(X)$.

###### Lemma

Let $f \colon X \to Y$ be a continuous function such that the image $f(X) \subset Y$ is a closed subset.

Then there is a homeomorphism

$Cone(f)/Cone(X) \simeq Y / f(X)$

between the quotient space (example ) of the mapping cone of $f$ (def. ) by the cone of $X$ (def. ) and the quotient space of $Y$ be the image of $X$.

###### Proof

Consider the following diagram in Top:

$\array{ f(X) &\longrightarrow& Y \\ \downarrow &(po)& \downarrow \\ f(Cone(X)) &\longrightarrow& Cone(f) \\ \downarrow &(po)& \downarrow \\ \ast &\longrightarrow& Cone(f)/Cone(X) }$

Here the top square is a pushout by example , while the bottom square is a pushout by def. . Hence the total rectangle is also a pushout, by the pasting law (prop. ). But that total rectangle is the defining pushout for $Y/f(X)$ (by example ). Hence the statement follows by the universal property of the pushout.

## Cofiber sequences

###### Proposition

Let $f \colon X \to Y$ be a continuous function with closed image $f(X) \subset Y$. Write $Cone(g)$ in

$\array{ X &\overset{f}{\longrightarrow}& Y &\overset{g}{\longrightarrow}& Cone(f) &\overset{}{\longrightarrow}& Cone(g) }$

for the mapping cone (spring) of the inclusion $g$ of $Y$ into the mapping cone of $f$.

Then the canonical quotient coprojection

$Cone(g) \to \Sigma X$

to the suspension of $X$ (example ) is a homotopy equivalence (def. ).

###### Proof

Since $g \colon Y \to Cone(f)$ is a closed Hurewicz cofibration (by remark ), lemma gives that

$Cone(g) \to Cone(f)/g(Y)$

is a homotopy equivalence. But then there is the following evident homeomorphism

$Cone(f)/g(Y) = (Y \cup_f Cone(X))/Y \simeq Cone(X)/X \simeq S X \,.$

(graphics taken from Muro 10)

Hence from every $f \colon X \to Y$ with closed image, we get long sequences

$\array{ X &\overset{f}{\longrightarrow}& Y &\overset{g}{\longrightarrow}& Cone(f) &\longrightarrow& Cone(g) \\ && && && \downarrow^{\mathrlap{\text{homotopy} \atop \text{equivalence}}} \\ && && && S X &\overset{S f}{\longrightarrow}& S Y &\longrightarrow& \cdots }$

## Pointed cofiber sequence

(…)

Last revised on July 4, 2017 at 14:22:39. See the history of this page for a list of all contributions to it.