A Simple Model of Sea-Surface Cooling under a Tropical Cyclone

Abstract

A major ocean response to tropical cyclone (TC) wind is the mixing of warm sea-surface water with cool subsurface water, which decreases the sea-surface temperature (SST). The decreased SST (δT) under the TC (rather than the cooled water in the wake after the storm has passed) modifies the storm’s intensity and is of interest to TC intensity studies. Here, the author shows that δT (non-dimensionalized by some reference temperature) is linearly related to Ψ, a dimensionless (nonlinear) function of TC and ocean parameters: the TC maximum wind, radius, and translation speed, as well as the ocean’s 26 °C and 20 °C isothermal depths (Z₂₆ and Z₂₀). The Ψ can be estimated from observations. The modelled δT is validated against sea-surface cooling observed by satellites, δT_o, for typhoons during the May–December 2015 period in the western North Pacific. The result yields a best-fit, linear relation between δT_o and Ψ that explains ~60% of the observed variance: r² ≈ 0.6 (99% confidence). Tests show that the cube of the TC maximum wind and the ocean’s Z₂₆ account for 46% and 7%, respectively, of the observed variance, indicating their predominant influence on TC-induced cooling. Contributions from other parameters are less but not negligible.

1. Introduction

Tropical cyclones (TC) are one of the most awesome natural phenomena on Earth [1]. A TC derives its energy from warm surface water in the tropics and subtropics, where from summer to fall the sea-surface temperature (SST) often exceeds ~26 °C, the minimum SST deemed necessary for storms to develop [2]. As the TC moves, the wind (maximum speed V_o ≳ 33 m/s) stirs the upper ocean and mixes warm surface water with cool subsurface water, reducing the SST under the storm as well as in its wake.

The surface conditions, including the SST, directly under the TC core, say, over an area scaled approximately with the square of the radius of maximum wind (r_mw ≈ 50 km), has the most impact on the storm’s intensity [3,4]. At a typical translation speed U_h ≈ 5 m/s [5], a TC is moving supercritically, as far as the ocean response is concerned [6], such that U_h/c > 1, where the ocean’s mode-1 phase speed c ≈ 3 m/s [7]. Moving at ~5 m/s, the direct and most violent impact on a fixed location of the sea from the TC core lasts for a few hours, generally less than the local inertial period of approximately 1 day. The drop in SST under the TC core, δT, is then caused more by vertical mixing rather than by upwelling [8]. (In this manuscript, I shall omit the contribution of surface enthalpy loss). In the wake, the SST may also be changed by upper-ocean convergences and divergences, as well as by mixing through the turbulence produced by the shears and the breaking of near-inertial internal waves [9,10]. However, as the TC has long passed, any wake cooling has little impact on intensity.

Numerical ocean models are widely used to estimate TC-induced sea-surface cooling (e.g., [11,12]), and some are coupled to atmospheric TC models (e.g., [13,14,15]). In this paper, an analytical formula is developed and validated against remote-sensing observations for the drop in SST under the TC core, δT. The simple model can be more easily coupled to an atmospheric TC model.

2. Methods

2.1. The Model

I consider sea-surface cooling caused only by the vertical mixing of the near-surface warmer with subsurface cooler ocean waters. As mentioned above, for supercritically translating TCs, the time spent by the TC at a particular sea-surface point is sufficiently short (~hours) such that horizontal variations do not play a significant role in the sea-surface cooling. Thus, cooling due to upwelling and near-inertial internal wave physics, for example, is negligible compared with vertical mixing under the TC core, as I explained previously [11,12,16,17,18]. The same vertical-mixing-only assumption for sea-surface cooling has been successfully used in TC simulations [13].

The model’s upper ocean consists of a near-surface Layer 1 of thickness h₁ and a lower Layer 2 of thickness h₂. An inert, deep layer of infinite thickness is assumed to underlie Layer 2, but it does not participate in the mixing process. I take h₁ = Z₂₆ (>0), the 26 °C isothermal depth, and choose h₂ to be the thickness between the 26 °C and 20 °C isotherms, i.e., h₂ = Z₂₀ − Z₂₆, where Z₂₀ (>0) is the 20 °C isothermal depth. The blue dots in Figure 1 show the TC locations in the western North Pacific where the maximum sustained speed V_o ≥ 33 m/s (i.e., Category 1 and above on the Saffir–Simpson scale https://www.nhc.noaa.gov/aboutsshws.php accessed on 2 January 2023), from May–December 2015. The TC data are from the IBTrACS dataset [19]. Shading and contours are h₁ and h₂ on the basis of the EN4 reanalysis data [20] (https://www.metoffice.gov.uk/hadobs/en4/download-en4-2-2.html accessed on 23 September 2022). (So that they can more precisely correspond to the chosen TC period, h₁ and h₂ are shown as weighted means with weights based on the monthly number of TC locations = (20, 2, 40, 33, 26, 29, 9, and 7) for 2015 (May, June, July, August, September, October, November, and December)). The choice of the 26 °C for h₁ seems reasonable since TCs rarely survive over SST < 26 °C. The 20 °C isotherm for h₂ is convenient (but quite arbitrary) since h₂ ≈ h₁ over most of the region (Figure 1). Previous studies indicate that TC-induced mixing can penetrate as deep as ~100 m below the sea surface (e.g., [21]).

Let Layer 1’s uniform temperature and corresponding density be T₁ and ρ₁ and those of Layer 2 be T₂ (<T₁) and ρ₂ (>ρ₁). Suppose that the two layers mix adiabatically into a single layer. By mass and heat conservation, the uniform density and temperature after mixing (subscript ‘mix’) are weighted averages of Layers 1 and 2:

ρ_mix = (ρ₂ h₂ + ρ₁ h₁)/(h₁ + h₂), T_mix = (T₂ h₂ + T₁ h₁)/(h₁ + h₂).

(1)

The change in SST is then:

δT = T_mix − T₁ = −ΔT [h₂/(h₁ + h₂)],

(2)

where ΔT = T₁ − T₂ (>0) and its corresponding Δρ = ρ₁ − ρ₂ (<0) measure the stratification of the initial 2 layers before mixing. The ΔT and, therefore, δT are quite arbitrary, and additional information is given below to relate them to the TC wind. This is so, even if Z₂₆ and Z₂₀ are known, since the T-profile is unspecified. For example, at a given location, the profile may be assumed nearly uniform in Layers 1 and 2: T₁ ≈ SST = 30 °C and T₂ ≈ 20 °C with a rapid transition across the 26 °C isotherm, yielding ΔT ≈ 10 °C. Then, δT ≈ −5 °C (h₁ ≈ h₂) would potentially be the largest possible SST drop at that location, no matter the physical cause. This suggests that given a slowly varying T-field, say, from monthly analysis (subscript ‘A’), one may make use of (2) to estimate a lower-bound δT_a by assuming that T_1A and T_2A are integral means of the T_A-profile:

δT_A = −(T_1A − T_2A) × [h₂/(h₁ + h₂)],

(3a)

$${\mathrm{T}}_{1\mathrm{A}}={\int }_{-{\mathrm{h}}_1}^0{\mathrm{T}}_{\mathrm{A}} \mathrm{dz}/{\mathrm{h}}_1, {\mathrm{T}}_{2\mathrm{A}}={\int }_{-{\mathrm{h}}_1-{\mathrm{h}}_2}^{-{\mathrm{h}}_1}{\mathrm{T}}_{\mathrm{A}}\mathrm{dz}/{\mathrm{h}}_2.$$

(3b)

TC wind imparts mechanical energy to the upper ocean and raises its potential energy. To calculate δT, we equate the raised potential energy PE (>0) to the wind work (WE) that produces the mixing. Thus, PE = PE|_mix − PE|_2layers, where:

$${\mathrm{PE}}_{2\mathrm{layers}}={\int }_{-{\mathrm{h}}_1}^0{\mathsf{\rho }}_1 \mathrm{g} \mathrm{z} \mathrm{dz}\prime +{\int }_{-{\mathrm{h}}_1-{\mathrm{h}}_2}^{-{\mathrm{h}}_1}{\mathsf{\rho }}_2 \mathrm{g} \mathrm{z} \mathrm{dz}\prime ,$$

(4)

$${\mathrm{PE}}_{\mathrm{mix}}={\int }_{-{\mathrm{h}}_1-{\mathrm{h}}_2}^0{\mathsf{\rho }}_{\mathrm{mix}}\mathrm{g} \mathrm{z} \mathrm{dz}\prime ,$$

(5)

and, therefore, after using (1) and some manipulation:

PE = −(g/2) ∆ρ h₂ h₁ = (g/2) α ρ_r ∆T h₂ h₁       (J/m²).

(6)

Here, ∆ρ and ΔT are assumed to be simply related by:

Δρ/ρ_r = −α ΔT,

(7)

where ρ_r is the reference seawater density ≈ 1025 kg/m³, and α = ($\partial \mathsf{\rho }/\partial T$)/ρ_o is the thermal expansion coefficient of seawater, ≈ 3 × 10⁻⁴ K⁻¹ near the sea surface, assuming SST ≈ 28 °C and salinity ≈ 35 psu.

The TC wind power at any point on the ocean is ρ_aC_dV³ in J/(m²∙s), the scalar product of the surface drag ρ_aC_d|V|V and TC wind V, neglecting the background wind and the ocean current. Here, ρ_a is the air density, V = |V| the TC wind speed, and C_d (= 2 × 10⁻³) is the empirical drag coefficient, which is assumed to be constant. At a point on the sea surface where the TC passes:

$$\mathrm{WE}= \mathsf{\gamma }{\int }_0^{2\mathrm{P}}{\mathsf{\rho }}_{\mathrm{a}}{\mathrm{C}}_{\mathrm{d}}{\mathrm{V}}^3\mathrm{dt} (\mathrm{J}/{\mathrm{m}}_2).$$

(8)

Here, γ is the mixing efficiency taking into account that only a fraction of the wind work goes into mixing the upper ocean (e.g., [22]). The integral spans from time t = 0 to 2P. The (0, P) is the time taken for the “front half” of the storm to arrive, while (P, 2P) is the time taken for the storm’s “back half” to leave the point, assuming symmetry about P. Thus, P = L/U_h, where L is a radial scale that measures the TC’s (half) size, and U_h is the storm’s translation speed. The point experiences the TC wind before the storm arrives [23], during (0, P). As the TC center nears the point, V becomes a little complicated. It strengthens to a maximum just outside the eye [1,2,3], rapidly falls to zero as the eye moves in and the TC center crosses over the point, and peaks again on the other side of the eye before diminishing to a weak background as the TC leaves. Empirical formulae are available to describe V (e.g., [24]). However, since V appears in the integral, its exact form is not crucial. I choose a simple rise-and-fall sinusoidal form:

V = V_o sin[πt/(2P)],      V_o = maximum wind,

(9)

thus ignoring the rapid wind change with two maxima as the TC center passes. Substitute this formula for V into (8), set WE = PE from (6), use (2) to replace ΔT with δT, and divide through by a reference seawater temperature T_r (≈28 °C) to render the result nondimensional:

$$\frac{\mathsf{\delta }\mathrm{T}}{{\mathrm{T}}_{\mathrm{r}}}=-\mathsf{\Psi }; \mathsf{\Psi }=\left[\left(\frac{8}{3\mathsf{\pi }}\right)\left(\frac{\mathrm{L}}{{\mathrm{U}}_{\mathrm{h}}}\right)\left(\frac{{\mathsf{\rho }}_{\mathrm{a}}}{{\mathsf{\rho }}_{\mathrm{o}}}\right)\left({\mathsf{\gamma }\mathrm{C}}_{\mathrm{d}}{\mathrm{V}}_{\mathrm{o}}{}^3\right)\right]/\left[{\mathsf{\alpha }\mathrm{T}}_{\mathrm{r}}\left(\frac{\mathrm{g}}{2}\right){\mathrm{h} \mathrm{h}}_1\right], 0\ge \mathsf{\delta }\mathrm{T}\ge \mathsf{\delta }{\mathrm{T}}_{\mathrm{A}},$$

(10)

where h = Z₂₀ = h₁ + h₂, and δT_A is the lower bound given by (3). Equation (10) is the required formula for the SST cooling produced by a TC with maximum wind V_o and radial size L, translating at a speed U_h over a sea-surface point where the 26 °C and 20 °C isothermal depths are h₁ and h₂. (The expression is the same as that given previously in [25], except for the additional multiplicative factor of 2 because the ‘L’ spans the TC diameter instead of the radius for SST cooling that does not feedback to the TC, as well as other details). The dimensionless function Ψ depends entirely on the TC and its underlying upper-ocean characteristics. The cooling is greater for a stronger, slower-moving, and/or larger TC, as well as where the local upper-ocean isotherms dome upward, i.e., h₁ and/or h₂ is thin, and vice versa.

2.2. Input Data

To test the model, the SST cooling δT from (10) is compared against satellite SST data at TC track locations (Figure 1). I chose 2015 (May through December) TCs in the western North Pacific, due in part to my familiarity with the region [25,26]. However, all the data described below are global and available from the early 1990s, and the procedure should apply to other ocean basins and periods.

The IBTrACS dataset’s (https://www.ncdc.noaa.gov/ibtracs/ accessed on 16 August 2022) six-hourly TC track locations (longitude and latitude) and maximum wind speed V_o from Joint Typhoon Warning Centre (JTWC) and Japan Meteorological Agency (JMA) were averaged, after multiplying the JTWC 1 min wind by a factor 0.88 to make it consistent with the JMA’s 10 min wind. Note that although the model cooling (Equation (10)) is calculated at the IBTrACS six-hourly track locations 00, 06, 12, and 18 UTC, satellite SST data are given daily only at (or very near) 12 UTC. To compare the two SST cooling, I simply subsampled the model cooling at the 12 UTC TC track locations. I consider only TCs of Category 1 and above, for which the lifetime maximum wind speeds exceed 33 m/s (i.e., typhoons; see http://www.weather.gov.hk/wxinfo/news/2009/20090318_appendix1e.pdf accessed on 30 January 2023) and only at TC track locations where V_o ≥ 33 m/s. The U_h was calculated from track locations using central differencing. For the radial scale, I let L = 2 × r_mw from JTWC (the only agency with the data), where the factor “2” was chosen to make the slope of the linear regression between modelled and observed δT, discussed below, approximately one. The factor in (10) merely changes the slope but not the r² of the regression. However, for the model to be physically plausible, the factor needs to be ≈O(1). Sensitivity tests using constant values of U_h and r_mw, ~5 m/s and 50 km averaged for all 2015 typhoons, will be presented. The mixing efficiency γ is set = 0.02, selected in [25] on the basis of tests on 2 TCs. However, the value is within the range reported for strong boundary stirring at high buoyancy Reynolds number [22,27,28,29], and I use the same value here. The C_d is set to 2 × 10⁻³, which falls in the range of values for TC winds (see references [30,31], where extensive citations are provided). There is some observational evidence that C_d may drop off to small values for major category (≥3) TC winds, stronger than approximately 50 m/s [32]. The scatter is large, however.

For the remaining quantities on the right-hand side of (10): h₁, h₂, and δT_A, I assume that they evolve slowly at a much longer time scale than the TC’s time. The assumption enables (10) to be useful since these ocean parameters can be readily estimated, and the δT is determined by the TC parameters alone: V_o, U_h, and L. I chose the EN4 version 4.2.2 monthly analysis (subscript ‘A’) of ocean temperature T_A, which is available from 1900 through the present [20]. The global gridded data is 1° × 1° with 42 levels. The top 18-level distribution, which includes the 20 °C isotherm, is z = −(5.02, 15.08, 25.16, 35.23, 45.45, 55.69, 66.04, 76.55, 87.27, 98.31, 109.81, 121.95, 135.03, 149.43, 165.73, 184.70, 207.43, and 235.39) m below the sea surface (z = 0 is sea surface). The monthly SST (denoted as “EN4 SST”) is assumed to be equal to the “T_A” at the first level z = −5.02 m; it will be used below to estimate the observed cooling from satellite SST. The Z₂₆ is calculated by linear interpolation to 26 °C of the temperatures at these depth levels, similarly for the Z₂₀. The T_1A and T_2A in (3) are then calculated using the trapezoidal integration rule. The condition δT ≥ δT_A was applied only 3% of the time when the modelled δT was outside the physically plausible cooling range.

2.3. Satellite SST Data

I use the daily (at 12 UTC), gridded global SST data produced by the Remote Sensing Systems, downloaded from https://data.remss.com/SST/daily/mw_ir/v05.1/netcdf/2015/ accessed on 3 October 2022. The REMSS SST incorporates microwave and infrared information. A useful description is given at https://podaac.jpl.nasa.gov/dataset/MW_IR_OI-REMSS-L4-GLOB-v5.0 accessed on 3 October 2022. The product is updated daily in near-real-time at a relatively high, 9 km resolution using optimal interpolation [33], with correlation scales of 3 days and 100 km. These relatively small temporal and spatial interpolation windows reduce, but do not eliminate, the influence of neighboring (past and future) SST information on the TC-induced cooling at a particular track point [11,12]. Nonetheless, compared with other gridded products I examined, the REMSS SST shows the least smoothing. The REMSS SST has been used in other TC studies [34,35]. To display TC-induced cooling, one customarily takes the difference of the gridded satellite SST at two different TC times: “later” minus “earlier”. The TC’s initial time is often taken as “early”, assuming that the corresponding SST represents the quiescent ocean at that time. The assumption breaks down when multiple TCs overlap, during an unusually active TC season, for example. This is the case for the TCs studied here in the western North Pacific preceding the very strong 2015/2016 El Nino [36]. Here, I set “early” = “EN4 SST”, interpolated to the TC’s initial time, and the observed cooling is then:

δT_O = REMSS SST − EN4 SST.

(11)

Since the EN4 SST is monthly in near-real-time, it approximates a quiescent background SST that evolves slowly compared with the fast SST changes caused by TCs (and/or other synoptic events).

3. Results

To illustrate how the modelled cooling δT (Equation (10)) compares against observation (δT_O, Equation (11)), Figure 2 shows an example of Typhoon Soudelor from 30 July to 11 August. The storm’s daily track points are superimposed on the EN4 SST and Z₂₆ plots for the same period. Using the TC and ocean information, the δT is then calculated at each point, as described in Section 2.2. I plot the modelled cooling as Gaussian contours with peak δT around each track point (x_tc,y_tc):

δT_con = δT(x_tc,y_tc) exp{−[(x − x_tc)² + (y − y_tc)²]/R²},     R = 200 km,

where (x,y) are (longitude, latitude); the contours are for graphical display only. Thus, the maximum cooling predicted by the model occurred at the 4 August track point (where δT_con = −2 °C can be seen; the peak δT = −2.35 °C). The model predicts little cooling over the region where the Z₂₆ is deepest ≈ 100~120 m, during the first half of the TC’s life (30 July–2 August), in good agreement with Soudelor’s intensity changes described in [26].

Figure 3 shows an example of δT_O on 7 August 2015, during Typhoon Soudelor. From a map such as this, the observed TC-induced cooling is defined as δT_O(x_tc,y_tc) = the minimum of δT_O (i.e., maximum cooling) found within the white circle trailing the track point (x_tc,y_tc). There are a total of 166 (x_tc,y_tc) track points from 17 TCs from May–December 2015 (Figure 1). These δT_O(x_tc,y_tc) are then compared against the modelled δT(x_tc,y_tc).

The above procedure was repeated for all TCs track points at 12 UTC. Figure 4 compares δT_O and δT, plotted as observed (Y-axis) vs. modelled (X-axis) cooling. To make the comparison, since the model predicts only the change in SST, δT, from some arbitrary background, I set δT = δT_O at the initial (x_tc,y_tc) point of each TC. This removes the initial bias. It effectively assumes that the initial point’s observed cooling when the TC wind is generally weak is caused by some background, large-scale cooling on that day. The initial bias is approximately −0.7 °C. Figure 4 is for the case where h₂ in (10) is set to its mean for all the track points (denoted by < >): <h₂> = 80.6 m. We will discuss other cases below.

Most points cluster around the regression line. The RMS difference between the model and observation, approximately 0.77 °C, is largely caused by several points lying far below the regression line, indicating that the model severely underestimates the observation at these points. However, they could also be an artifact of the optimal interpolation analysis used for the satellite SST, which uses data from the future and past, as well as from the neighboring points, as mentioned before.

The regression line is

δT_p = −a T_r Ψ + b,       a = 0.94,     b = −0.09 °C,

(12)

at the 99.9% confidence, and with the explained variance r² ≈ 0.6. The predicted cooling, δT_p, depends on the dimensionless Ψ, which is fully determined once the TC’s peak wind (V_o), size (r_mw), and translation speed (U_h), as well as the upper-ocean warm depths, Z₂₆ and Z₂₀, are known or can be estimated.

We can assess the influence of these parameters on the modelled surface cooling by fixing each of them, in turn, while leaving others to vary and then repeating the regression analysis (

Table 1. Model sensitivity to TC and ocean parameters (see text for details). All r² and slope “a” values are significant at the 99.9% confidence level, except for the fixed V_o³ case (italicized numbers).

	All Vary	Vo3 Fixed	h1 Fixed (Z26)	h2 Fixed (Figure 4)	h Fixed (Z20)	rmw Fixed	Uh Fixed	rmw/Uh Fixed	Vo3 Varies	Vo3 and h1 Vary
Mean	-	49,375 (m/s)3	74.5 m	80.6 m	155.1 m	48.8 km	5.0 m/s	2.7 h	-	-
r2	0.57	0.05	0.38	0.63	0.60	0.57	0.55	0.53	0.46	0.53
a (slope)	0.92	0.32	0.88	0.94	0.98	0.77	0.90	0.73	0.76	0.73
RMS (Model—Obs) °C	0.84	2.01	1.02	0.77	0.80	0.96	0.86	1.05	1.04	1.05

; Figure 5). I choose the parameter’s mean as the fixed value. Column 2 lists the original model when all of the parameters are variable; it is the reference case for comparison. Columns 2–9 show the results of the six parameters: V_o³, h₁, h₂, h, r_mw, U_h, and r_mw/U_h each fixed, in turn, at their mean values, as shown in Row 2.

Wind power ($\propto$V_o³) has the most impact on δT, as can be expected since wind provides the energy for water mixing that cools the sea surface. Fixing it reduces the r² to near zero, eradicating any model predictability.

The h₁ (i.e., Z₂₆) has a strong effect on δT. Fixing it reduces the explained variance by 19%, to r² = 0.38. This agrees with various studies that emphasize the importance of upper-ocean warm layers on SST cooling and TC intensity change (e.g., [13,18,25,26] and references cited therein). The RMS difference between the model and observation also deteriorates (becomes 15% larger).

Fixing the h₂ improves the explained variance to r² = 0.63, the slope to a = 0.94, and reduces the RMS difference to 0.77 °C. In addition to the uncertainty in the observations deeper below the surface, this test also reflects the model’s uncertainty in choosing Z₂₀ as the second mixing layer.

Fixing the h (i.e., Z₂₀) also improves the explained variance to r² = 0.60. It improves the slope to 0.98 (i.e., closer to 1) and reduces the RMS difference.

Fixing the TC’s size ($\propto$r_mw) has little effect on the explained variance; the resulting r² = 0.57 is unchanged from the reference case. This is because (1) r_mw appears as a linear term in the model (see Equation (10)); and (2) once a TC has matured, the r_mw remains at a fairly constant value (approximately 20–30 km, somewhat less than the mean). Thus, since the δT is heavily weighted by strong V_o³, its dependence on r_mw is a simple linear scaling that changes the slope of the regression with little effect on the variance, i.e., on r².

Fixing the TC’s translation speed (U_h) reduces the explained variance by 2%, to r² = 0.55.

Fixing the time spent by the TC over the sea surface under the core (r_mw/U_h) reduces the explained variance by 4%, to r² = 0.53.

It is of interest to test how much variance can still be captured by retaining only V_o³, as well as V_o³ and Z₂₆. The results are given in the last two columns of Table 1 and Figure 5. Wind power alone (V_o³) alone accounts for 46% of the TC-induced cooling. Including a variable upper-ocean warm layer Z₂₆ improves the explained variance by another 7%. Compared with the complete model (the “All Vary” case), including the other variables adds only another 4% to the explained variance. However, keeping the second layer h₂ fixed improves the explained variance by 10% to 63%.

4. Discussion

Despite its simplicity, the model explains more than 50% of the variance of TC-induced cooling under the storm’s core. Tropical cyclone wind produces upper-ocean mixing and is necessarily the main source of the variance. That is unsurprising. However, the model also shows that TC wind-induced mixing is modulated by changes in the upper-ocean warm layer, in particular the Z₂₆, as the storm travels over the sea surface, which further contributes to explaining the TC-induced cooling. As cited above, several studies have described the potential effects of the upper-ocean warm layer on TC-induced cooling and TC intensity change. For example, TC travelling over a warm eddy, where the Z₂₆ is thick, would be shielded from the sea-surface cooling, and therefore, the storm could potentially more easily further intensify. On the other hand, TC travelling over a cold eddy where the Z₂₆ is thin would tend to weaken due to its exposure to potentially greater surface cooling. The underlying theory has not been clarified, however, to the best of my knowledge. Equation (10) shows that the inverse relationship between surface cooling and upper-ocean layers, i.e., less (more) cooling for thicker (thinner) upper-ocean warm layers, is because the drop in SST varies like (Z₂₀ Z₂₆)⁻¹. The model predicts a singularity, i.e., infinite cooling as Z₂₆ ~ 0. On the basin scale, the Z₂₆ ≈ 0 occurs at a latitude of approximately 27°~35° depending on the month of the year (see, e.g., Figure 1), which is consistent with the observed fact that a TC can less easily intensify poleward of that latitude (e.g., [2]).

The model can be modified and improved in several ways. The model can include the additional effect of sea-surface cooling by latent and sensible heat losses. Effects of ocean mesoscale eddies can be included through Z₂₆ estimated from satellite sea-surface height anomaly field. The anomaly can be assumed as “frozen” as the TC passes since the TC time scale is much faster than the time scales of the response of the eddy to the storm. Finally, the model can be coupled to an atmospheric TC model.