a. Water balance equations

To formulate the underlying water balance equations, we express all moisture fluxes between the boxes as functions of the system’s moisture state. For atmospheric boxes, the moisture state is given by the mean water vapor path w (mm), and for the land box by the unitless mean relative soil moisture saturation s. Since the ocean does not dry out, it does not require a moisture variable. Hence, the full information on the moisture state of the land–ocean–atmosphere system at any moment in time t is given by the set of state variables $\{w_o(t),w_{\mathcal{l}}(t),s(t)\}$.

b. Empirical relationships for water fluxes

c. Parameter ranges

We want to constrain χ for our system in equilibrium and test its sensitivity to parameter choices. To do that, we need to define plausible ranges for all free model parameters, thereby constructing our search space. The chosen ranges are summarized in Table 1.

Table 1

Free parameters for the closed model simulations with uniform random sampling of parameter values.

View Table

The ranges for s_pwp and s_fc are taken from data for different soil types presented in Hagemann and Stacke (2015), where we discard the extreme cases of pure sand and peat. After converting the provided volumetric data to relative soil moisture saturation values, s_pwp ranges between 0.15 and 0.55. We notice the fairly consistent relationship, s_fc = s_pwp + 0.3, and use it to reduce the number of free model parameters by one. Entekhabi et al. (1992) provide values for e_p, nz_r, r, and ϵ for both a semihumid and a semiarid climate. We take the values from these two climates as limits for the respective parameter ranges and vary e_p between 4 and 6 mm day⁻¹ and nz_r between 50 and 120 mm. Entekhabi et al. (1992) set r = 6 for both cases but we let the runoff exponent range between 2 and 6, motivated by Rodriguez-Iturbe et al. (1991) who used r = 2 in an illustrative example. Since both studies agree on ϵ = 1, we vary this parameter only slightly between 0.9 and 1.1. Note that ϵ values larger than one could lead to unphysical solutions where runoff exceeds precipitation. Such unphysical cases must be excluded from the analysis, but it turns out they never occur. The precipitation parameters are taken from Bretherton et al. (2004) (where our b is called r). We use their fitting parameter values for monthly and daily data as bounds for a and b. Bretherton et al. (2004) also give a typical value, w_sat = 72 mm, for regions of tropical convection and we deem it appropriate to vary w_sat between 65 and 80 mm. Ocean evaporation can be constrained from observations. For instance, Kumar et al. (2017) found the mean evaporation from tropical ocean surfaces to range between 2.5 and 2.8 mm day⁻¹, while an earlier study by Zhang and McPhaden (1995) indicates higher values of about 3.5 mm day⁻¹. Based on these findings, we let e_o range between 2.5 and 3.5 mm day⁻¹. This range also encompasses 3 mm day⁻¹—the mean value of precipitation over both tropical land and ocean. Last, τ is constrained by computing the smallest and largest value of u/L, respectively, where we assume plausible wind speeds in the lower troposphere between 1 and 10 m s⁻¹, and let the domain length vary between 1000 and 40 000 km, the upper limit corresponding to Earth’s equatorial circumference. Thus, we obtain a range for τ between about 0.002 and 0.864 day⁻¹.

d. Model assumptions

The simplicity of the proposed model owes to a number of assumptions, some of which are important to be made explicit. Foremost, we assume that each model box has well-mixed physical properties so that all interactions are adequately described in terms of spatial mean quantities. Second, we determine the system’s moisture state by water balance equations only, ignoring the potential effects of energy balance considerations such as the influence of a diurnal cycle. Energetic conditions are kept constant and only enter indirectly through the values of energy-dependent parameters such as e_o or w_sat. Third, we prescribe a background wind speed with only a horizontal and constant component. Last, and more importantly, we assume that the functional relationship between precipitation and water vapor path from Eq. (6) holds over both land and ocean with the same choice of parameter values. By comparing the range of obtained χ values from our simple model to known values from the real world, we will discuss in section 6a what we can conclude about the potential processes that control the land–sea precipitation contrast in the real world.

a. Atmospheric transport parameter τ

The relationship between χ and τ in Fig. 5a is strongly nonlinear and leads to variations of the χ mean between 0.5 and 0.98, confirming the high sensitivity determined in the mutual information analysis. Physically, τ corresponds to the fraction of the domain length that moisture can travel horizontally in the atmosphere in one day. As such, it can be interpreted as the efficiency of atmospheric moisture transport or efficiency of horizontal mixing: The higher the value of τ, the more efficient the mixing, and the lower the moisture differences, $\mathrm{\Delta }w=w_o-w_{\mathcal{l}}$, between the two atmospheres. Since we link precipitation and atmospheric moisture through the same parameterization P(w) over land and ocean, a small Δw implies similar precipitation rates over land and ocean and a χ value close to one. In contrast, inefficient mixing due to small values of τ can sustain larger moisture differences between land and ocean and leads to smaller values of χ. This theoretical control of τ on χ is confirmed in Fig. 5a where χ tends to be small for low values of τ and converges toward one as τ increases. The spread around the τ mean is reduced as τ increases, making τ a better predictor for χ, the larger its value. Figure 5a also suggests that τ has a strong control on the lower bound of the spread around the χ mean which increases with τ. In contrast, the upper bound of the spread, χ = 1, seems independent of τ.

We can understand how τ influences the lower bound of the spread of χ values by returning to the advection equations (8) and (9), each of which can be rephrased as the product of Δw and an advection efficiency, namely, τ/α for land advection and τ/(1 − α) for ocean advection. Only if both advection efficiencies are low, a large moisture difference and, hence, small χ can be sustained. The land fraction affects ocean and land advection efficiencies in opposite ways, suggesting a minimum of overall advection efficiency for intermediate α. If advection was the only process at play, this minimum would be located at α = 0.5, and it would be left to τ to set the final value of the lowest possible advection efficiency, and thereby the smallest possible χ for that value of τ. In the following, we will see that the complexity of land–atmosphere interactions adds further parameter controls on the lower bound of χ and leads to an asymmetry in the relationship between χ and α such that the lowest χ value is found at a larger land fraction than α = 0.5.

b. Land fraction α

Figure 5b illustrates how α impacts the value of χ. The χ mean varies between 0.91 and 1.0 on a u-shaped line between the extreme cases of an ocean-only, α = 0, and land-only, α = 1, scenario where in both cases χ is close to one. As for τ, the impact of land fraction changes on the value of χ is rooted in their control on the advection efficiencies, τ/α and τ/(1 − α). The particular role of α is to differentiate between the efficiency of moisture export out of the ocean subdomain and the efficiency of moisture import into the land subdomain by setting the different areas over which the advected moisture gets distributed. If α is small, the imported moisture gets distributed over a small land size, making land advection efficient and the rate per unit area high. The reverse applies for a large land fraction. In either case, mixing between the two atmospheres is efficient and the smaller atmosphere adopts the moisture conditions of the larger one, resulting in a χ value close to one.

While the model cannot handle the exact endpoints of the α range, we can examine them in a thought experiment: imagine a domain fully covered by ocean. Water balance would require the ocean precipitation to balance ocean evaporation, P_o = E_o. If we now introduced an infinitesimal patch of land, some of the evaporated moisture would be advected into the tiny land atmosphere without significantly altering the moisture conditions over the vast ocean. Since τ/α is high, the atmospheric conditions over land would rapidly converge to those over the ocean. Hence, for such small α, the system is expected to behave as if the land did not exist. Atmospheric conditions would be overall moist with nearly the same land and ocean precipitation rates close to E_o, leading to a χ of one. At the other extreme, imagine a pure land domain but with the assumption that runoff to some external reservoir remains possible. The runoff would continuously reduce the soil moisture saturation and with it evapotranspiration and precipitation until the trivial equilibrium solution $\{s=0,w_{\mathcal{l}}=0\}$ is reached. Allowing for advection from an infinitesimal ocean would not change the picture much. With high efficiency τ/(1 − α), almost the entire but nevertheless small amount of evaporated oceanic moisture would be exported to the land, leaving behind a fairly dry ocean atmosphere and hardly affecting the dry state of the large land atmosphere and soil. Hence, the system would behave as if the ocean did not exist with similarly low precipitation rates close to zero in both subdomains.

The transition between the ocean-only and land-only scenario is best understood by examining the individual rain rates P_o and $P_{\mathcal{l}}$ over the range of α as shown in Fig. 6. The two extremes are connected by a regime of monotonic drying as the ocean surface—the only true source of moisture for the system’s hydrological cycle—shrinks. Although not shown, the overall reduction of available moisture as the land fraction increases also manifests itself in a soil moisture decrease. We can investigate the shapes of $P_{\mathcal{l}}$ and P_o theoretically, assuming all parameters except α to be fixed. Expressions for the rain rates are found by imposing equilibrium conditions on Eqs. (2) and (3), yielding

$P_{\mathcal{l}}=E_{\mathcal{l}}[s(\alpha )]+\mathrm{\Delta }w(\alpha )\frac{\tau }{\alpha },$(13)

$P_o=e_o-\mathrm{\Delta }w(\alpha )\frac{\tau }{1-\alpha }.$(14)

Note that s(α) and Δw(α) are implicit functions of the land fraction but that we lack analytical expressions for them. As α increases in Fig. 6, $P_{\mathcal{l}}$ decreases more strongly than P_o because land precipitation is not only affected by (initially sharply) decreasing 1/α but in addition by a reduction in evapotranspiration as s declines. In contrast, ocean precipitation is only reduced by (initially weakly) increasing advection. The additional influence of α on $P_{\mathcal{l}}$ through $E_{\mathcal{l}}$ causes the asymmetry of the χ(α) mean in Fig. 5b with a minimum around α = 0.75 instead of 0.5. Although χ(α) is expected to be u shaped for every combination of the other parameter values, the exact location of the minimum will differ. Generally, the width of the spread of χ values indicates that the land fraction is the dominant predictor for χ near its extreme values but that its influence weakens toward intermediate values.

View Full Size

Ocean precipitation rate P_o and land precipitation rate $P_{\mathcal{l}}$ vs land fraction α. The white lines show χ mean values computed for 100 bins along the α axis. The transition from an ocean-only to a land-only model scenario is marked by a decrease of both precipitation fluxes, indicating overall moist conditions for small land fractions and dry conditions for large land fractions.

Citation: Journal of Hydrometeorology 24, 6; 10.1175/JHM-D-22-0162.1

• Download Figure
• Download figure as PowerPoint slide

With the understanding of the influence of α developed in this section, we can now explain the first two seemingly unphysical behaviors in Fig. 3, as described at the end of section 4. They concern the fact that there is hardly any rain over the ocean when the soil is dry and that ocean advection tends to decrease with increasing s while land advection increases. The model runs which populate the regime of low soil moisture in Fig. 3 share the property of large land fractions and therefore equilibrate to overall dry conditions, both over land and over ocean. Both atmospheres can consequently only yield very little rain. The amount of moisture exchanged between land and ocean translates to an advection rate per unit area that is high for the small ocean and low for the large land. The opposite situation is found at the largest s values in Fig. 3, which originate from model runs with very small α. There, $A_{\mathcal{l}}$ is at its maximum while A_o nearly vanishes. In the intermediate s-regime, values of α are intermediate and have little influence on the shape of the fluxes.

c. Permanent wilting point s_pwp and field capacity s_fc

Variations in the soil parameters lead to mean variations of χ similar to the effects of land fraction changes, with the mean varying between χ = 0.88 and 0.98 for both the permanent wilting point (Fig. 5c) and the runoff exponent (Fig. 5d). In Fig. 5c, χ shows an almost linear decrease for an increase in s_pwp. To understand this behavior, another thought experiment is helpful: Let us consider a system in equilibrium for some value of the permanent wilting point, e.g., s_pwp = 0.3, and examine how the system would respond if this value was suddenly changed to s_pwp = 0.4, as illustrated in Fig. 7. The presented arguments assume that the other parameter values stay fixed when varying s_pwp but we can see from the right panel of Fig. 7 that the influence of the permanent wilting point also leaves its imprint on the mean soil moisture state (white line) in the form of a clear increase with s_pwp. We therefore make use of the mean value for illustration purposes. The left panel of Fig. 7 depicts the evapotranspiration curves for s_pwp = 0.3 and 0.4, respectively, with fixed e_p = 5.0 mm day⁻¹. Because s_pwp and s_fc are equidistant for different soil types, a change of s_pwp merely shifts the evapotranspiration curve along s.

View Full Size

Influence of an increase in s_pwp on the equilibrium state. (left) Higher values of s_pwp shift the graph of the $E_{\mathcal{l}}$ parameterization toward larger s. (right) Equilibrium values of the soil moisture saturation from CM data plotted against s_pwp values. The blue dots mark the initial equilibrium state, the green dot represents a temporary state of soil moisture imbalance due to a sudden increase of s_pwp, and the orange dot marks the eventually attained equilibrium state for the new s_pwp value.

Citation: Journal of Hydrometeorology 24, 6; 10.1175/JHM-D-22-0162.1

• Download Figure
• Download figure as PowerPoint slide

The mean soil moisture value in the CM data for s_pwp = 0.3 is s = 0.45. This initial state for our thought experiment and its corresponding evapotranspiration value are displayed as blue dots in Fig. 7. An abrupt increase of the permanent wilting point to s_pwp = 0.4 would lead to the following sequence of events. First, evapotranspiration experiences an instantaneous drop ΔE_inst (first red arrow connecting the blue and green dots). The green dot represents a temporary state where the model is not in equilibrium because the soil receives more moisture from precipitation than it loses through evapotranspiration and runoff. Consequently, the soil moistens. As time progresses, the system attains a new equilibrium state (orange dot) at a higher s value. However, as the soil moistens, not only evapotranspiration but also runoff increases so that the soil does not moisten enough to reach the initial $E_{\mathcal{l}}$ flux. As a consequence, the land atmosphere becomes drier and land advection increases. Since the ocean atmosphere needs to supply more moisture to the land, w_o decreases. Therefore, the increase in advection from increased Δw, needed to balance the increased runoff, is only possible if $w_{\mathcal{l}}$ decreases more strongly than w_o. Accordingly, $P_{\mathcal{l}}$ decreases more strongly than P_o, which explains why χ declines with increasing s_pwp.

In effect, the new equilibrium for a larger s_pwp value would be characterized by a moister soil with larger runoff but reduced precipitation, thus leaving less moisture to evapotranspiration. The new $E_{\mathcal{l}}$ would be ΔE_eq smaller than its initial value. This response to an increase in s_pwp is also responsible for the third seemingly unphysical behavior, namely, the decline of evapotranspiration for intermediate s values previously seen in Fig. 3. Precipitation and evapotranspiration are reduced because states with moister soils correspond to simulations with larger s_pwp.

d. Runoff exponent r

The relationship between χ and r in Fig. 5d resembles χ(s_pwp) but with opposite trend: χ increases with r while it decreases with s_pwp. Indeed, the similarity originates from a similar physical mechanism. In the formulation of the runoff fraction, R_f = ϵs^r, r enters as the exponent. As for the s_pwp dependence, we can conduct a thought experiment, starting from a system in equilibrium which then responds to a sudden increase of r. Since s has values between zero and one, an increase of the runoff exponent reduces the value of R_f and soil moisture increases. As a consequence, also both atmospheres start to moisten: First, $w_{\mathcal{l}}$ increases through increased evapotranspiration from the wetter soil, thereby reducing the moisture difference between land and ocean. Second, the reduction of Δw implies reduced advection with the effect that the ocean atmosphere retains more of its moisture and w_o increases. Eventually, the decreasing advection matches the runoff which started to intensify again after the initial drop. A new equilibrium is attained in which runoff is reduced compared to the initial state with lower r value. Equally reduced land advection with lower Δw reflects more similar atmospheric moisture conditions and, hence, increased χ.

Equilibrium states with higher r value also have stronger precipitation rates from higher $w_{\mathcal{l}}$ and w_o. In Fig. 3, we see the influence of r in both the low and high soil moisture regime, where it works in tandem with the land fraction to shape the moisture fluxes. Both precipitation increases for low and high s, respectively, correspond to a combination of increasing r and decreasing α. In these regimes, the combined influence of r and α dominates over the influence of s_pwp discussed earlier.

a. Relaxing assumptions of the closed model

To begin with, we started out with the hypothesis that precipitation is neither favored by land, nor ocean, meaning that different surface characteristics do not affect the way it rains: for the same water vapor path, the model computes the same amount of rain over land and ocean. In reality, while we can generally expect a moister atmosphere to yield more rainfall, the shape of P(w), determined by the values for the a and b parameters, could differ over different surfaces. With its tendency for too low χ values, our model suggests that our initial hypothesis about P(w) was wrong and that in reality, precipitation is actually favored over land. In the model, we can favor precipitation over land most easily by choosing a smaller b parameter over land than over ocean, leading to higher precipitation rates over land than over ocean for the same water vapor path. As a proof of concept, we ran model simulations with different choices for b over land and ocean. When choosing b just 0.1% smaller over land than over ocean, already about 20% of the simulations yielded a χ value larger than one. That it rains differently over land and ocean would be in line with the conclusions of Hohenegger and Stevens (2022), who found more precipitation over tropical land than what is expected based on the tropical land–sea distribution and rainbelt position, and also with the results of Ahmed and Schumacher (2017), who found distinct differences in the P(w) relationships over land and ocean based on observations. Further studies examining differences in P(w) over different surface types, and constraining the realistic ranges of the a and b parameters are needed to conclude whether precipitation processes are responsible for the higher χ values found in nature.

Second, we treat all model boxes as being homogeneous and well-mixed which allows us to work with mean fluxes rather than resolving the horizontal direction explicitly. It is well known, however, that an airstream traversing an oceanic region will moisten along its trajectory since mean ocean evaporation typically exceeds mean ocean precipitation and the reverse applies to land regions. Furthermore, Ogino et al. (2016) and Ogino et al. (2017) found that the conventional view in which Earth’s surface gets divided into ocean and land misses out on particular interactions driven by the land–sea contrast which are confined to a coastal region, a few hundred kilometers seaward and landward from the coast. These coastal regions receive more rain than both the open ocean and inland continental regions. Also, Bergemann and Jakob (2016) found that tropical rainfall over land associated with coastal effects such as sea breezes can occur under drier atmospheric conditions than rainfall over the open ocean. Hence, adding coastal zones with a specific coastal precipitation parameterization to the model is another flavor of the argument that—for precipitation enhancement over land—it has to rain differently in different subdomains. Coastal zones might also capture the fact that precipitation enhancement is particularly strong over relatively small landmasses where coastal effects are expected to be more influential.

Third, the model has neither an energy budget, nor a diurnal cycle. As a consequence, phenomena associated with pressure gradient forcing like diurnal sea breezes which tend to enhance precipitation over land are not captured. Energy dependence and a diurnal cycle can be implemented in different ways—either fundamentally by coupling water balance and energy balance equations, or indirectly by introducing a diurnal cycle in energy-dependent parameters such as τ, e_p, or w_sat. Even with the same P(w) relationship across the domain, a diurnal cycle may lead to χ > 1 through two pathways: evapotranspiration might be enhanced more strongly than ocean evaporation during the day. At the same time, the wind field would become variable and might exhibit convergence over land, potentially leading to upgradient moisture transport which was formerly disabled due to the assumption of a constant background wind speed. The combined effect of these two pathways may lead to a high enough concentration of moisture over land during the day to yield higher temporary rain rates over land than over ocean. If the reverse transport during the night does not fully compensate for the daytime precipitation signal, then χ might be larger than one on average. In other words, the diurnal cycle of available energy may explain why χ is close to one and can even be larger than one in reality.

Last, our model configuration with one land and one ocean box is not representative of the real tropics. It is likely that a different land distribution with more boxes would lead to higher χ values because smaller box sizes generally increase the advection efficiencies. If, in addition, the boxes were differently sized, we might see instances where χ > 1. In equilibrium, each ocean atmosphere would still be moister than the leeward land atmosphere, but it cannot be precluded that weighting the precipitation rates by the different box sizes would yield a stronger mean land than ocean precipitation.

b. Opening the closed model

The previous arguments still treat the land–ocean–atmosphere system as a closed model. This assumption may be valid over the full tropics, assuming a negligible net moisture exchange with the extratropics, but it is certainly invalid over islands, where land precipitation enhancement is typically observed. Hence, allowing for atmospheric inflow and outflow out of the domain might create χ values larger than one. We test this hypothesis in two ways.

First, moisture import or export from an external environment outside the model boundaries can be incorporated by an additional advection term, A_ext (mm day⁻¹) in Eq. (3) for oceanic water vapor. A positive A_ext denotes inflow of external moisture, while a negative A_ext means that the ocean atmosphere loses moisture through the model boundaries. This construction mimics the case of an island surrounded by an ocean under the influence of large-scale convergence or divergence. But is this change in the model framework sufficient to create scenarios for which χ > 1?

We argue that the answer is no: regardless of whether the system is gaining or losing moisture through its boundaries, an equilibrium state still requires a net transport of moisture from ocean to land, and therefore $w_o>w_{\mathcal{l}}$ with χ < 1. The term A_ext acts in a similar fashion as ocean evaporation. Under the influence of moisture convergence, the positive A_ext is equivalent to an increase in E_o and merely increases the moisture content in all boxes. In the case of moisture divergence, negative A_ext acts like a reduction of E_o and an overall drier equilibrium state is attained as long as |A_ext| < E_o. If the loss through the model boundary is stronger than the moisture input from the ocean surface, the system will undergo drying until the trivial solution $\{w_o=0,w_{\mathcal{l}}=0,s=0\}$ is reached.

Second, we can also open the model by allowing moisture to enter the domain from one side and leave it on the other side. As an illustrative example for this type of open model, we consider the simplest configuration with a land box of length $L_{\mathcal{l}}$, placed between two equally big ocean boxes of lengths L_o1 = L_o2. The subscript o1 refers to the ocean in front of the island as seen by the airflow horizontally traversing the domain at constant wind speed, while o2 refers to the ocean behind the island. The equations governing the evolution of w_o1, $w_{\mathcal{l}}$, and w_o2 and s are formulated in analogy to the closed model equations and can be found in the appendix along with a model sketch. This open model requires one additional parameter, the boundary water vapor path w₀ with values between 0 mm and w_sat, which denotes the water vapor path at the windward model boundary. It enters the model equations in the advection term for the first ocean. As for the closed model, we perform 100 000 simulations with randomly chosen combinations of parameter values from Table 1 with the modification that the full domain length is varied between 200 and 2000 km. The obtained set of equilibrium solutions is referred to as “OM data.”

We perform the same sensitivity analysis as for the closed model to understand which parameter combinations lead to χ values larger than one, and to which parameters χ is most sensitive. Opening the model does not fundamentally change the principal sensitivities but modifies their order of importance: The most sensitive parameter is now w₀ with I_MI(χ, w₀) = 254, followed by α with I_MI(χ, α) = 202 and τ with I_MI(χ, τ) = 147. All other parameters, including the formerly relevant soil parameters, have I_MI values lower than 8 and can be neglected as predictors for χ. To understand which parameter combinations lead to χ > 1, Fig. 8 shows PDFs of the values of w₀, α, and τ for simulations with χ values smaller and larger than one in blue and orange, respectively. Even though the distributions have a significant overlap, χ > 1 requires a large enough boundary water vapor path $w_0\gtrsim 38\text{mm}$ and becomes more likely with smaller land fractions. States with χ > 1 do not exist for α > 0.93 and are most likely around α = 0.05. In contrast, a value of χ < 1 seems to be possible with any value for α. Further inspection of the state variable values (not shown) reveals that equilibrium states with χ > 1 are overall moist. Soil moisture saturation values cluster close to and beyond the field capacity. Water vapor path values peak around 48 mm over both land and ocean and never get smaller than 38 mm. Simulations with χ < 1, in contrast, exhibit equilibrium states across the entire moisture spectrum, including very dry states with soil moisture values below the permanent wilting point.

View Full Size

Probability density functions of parameter values for w₀, α, and τ. Blue and orange graphs contain data from simulations with χ values smaller and larger than one, respectively. Ratios larger than one are only found for 6.7% of all simulations.

Citation: Journal of Hydrometeorology 24, 6; 10.1175/JHM-D-22-0162.1

• Download Figure
• Download figure as PowerPoint slide

As in the case of the closed model, the open model results are subject to the choice of the land distribution and may change for different numbers of land boxes. In addition, the open model is sensitive to the location of the land box which affects the relative size of the two ocean boxes. However, a test of different asymmetric configurations indicates that the presented statements are only affected quantitatively, not qualitatively.