3.2.1 Response to ∇p_a.

First let us look at the barotropic (n = 0) response of the idealized model to the pressure anomaly of the moving TC. For an equilibrium solution following the moving TC, where γ = −C_tc∂_x, Eq (2) gives because , , and (note Eq. (S8) in S1 Text, , and ψ₀(0) = 1 for the last equality). Here L is the characteristic scale of horizontal variation in π_a, which is according to S1 Fig in S1 Text, whereas is the barotropic radius of deformation. Considering that both π_a and π₀ vanish at infinity and that only the gradients of π_a enter the basic equations (S3) in S1 Text, we conclude

That is indeed the case as shown in Fig 3: The upper panels plot −π_a + const. and π₀ and the two curves nearly coincide; the difference between the two fields are very small (lower panel). Moreover, since ∇π + ∇π_a ≈ 0, u and v are almost zero (see Eqs. S8 in S1 Text) for the equilibrium response.

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Fig 3. p_a and ρ_oπ₀; the former is the pressure anomaly of the TC and the latter is a temporal average of the pressure response of the n = 0 RGM to p_a; ρ_oπ₀ is the temporal average in the moving coordinates over the last 3 d of the integration to suppress gravity waves.

The center of the TC is set at (x, y) = (5000 km, 3000 km) in the moving coordinates. The pressure anomaly is a sum of three TC instances 7000 km apart in the x direction to account for the zonal cyclicity of the computational domain. See S1 Text. But the result is not much different if only one instance is used (not shown) because the zonal width 7000 km is significantly larger than the zonal scale of the TC. (A) Zonal and meridional sections of −p_a + const. (blue dashed) and ρ_oπ₀ (orange solid) across the center of the TC, but the two curves are so close to each other that they almost coincide. p_a is shifted by the constant value 19.1hPa so that the curves coincide. See S1 Text. (B) ρ_oπ₀ + p_a − const., showing the residual.

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Baroclinic modes are negligible. For example, we have run the mode 1 RGM. The maximum amplitude of would be 4.8 hPa (not shown) if the pressure coupling coefficient, , were as large as . Even this hypothetical amplitude is significantly smaller than that of (Fig 3). It is actually much smaller because whereas . (See Eqs. (S8) and S1 Table in S1 Text and note that according to Section S1.2.6 in S1 Text). The actual amplitude is therefore like 4.8 × 0.0002 ≈ 0.001 hPa for the first baroclinic mode.

With neglect of baroclinic modes, the 3-d distribution of the barotropic pressure is π₀(x, y)ψ₀(z), but since ψ₀(z) = 1, the anomalous pressure is, at all depth, which is uniform in space (See Eq. S2c in S1 Text). How uniform it is is shown in Fig 3B, which plots the residual:p′ + p_a − const. The size of the residual is ∼0.2 hPa or less. The cause of this residual is not clear. It may be due to the gravity waves which may still remain after the temporal average because the data interval, 12 500 s, is not frequent enough or because the averaging period is not long enough; or it may be due to the approximation that led to the conclusion that ∇²π₀ ≈ −∇²π_a.

Fig 4 shows the w response. (Despite the 3-day average, it includes low-amplitude small-scale noise, which is likely due to the weakness of any dissipation in the numerical model. See Section S1.2.4 in S1 Text). Equations (S8f) and (S8g) in S1 Text give, on the stationary coordinates, because and π₀ = gη₀ (Eq S8g) in S1 Text. In the equilibrium state on the moving coordinate system, therefore,

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Fig 4. w₀ of the n = 0 RGM forced by p_a; average over the last 3 d of the integration to suppress gravity waves.

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On the front side of the TC, sea level is raised by the low atmospheric pressure (∂_xη₀ < 0 and ∂_xp_a > 0), which requires upward vertical velocity w₀ = −C_tc∂_xη₀ > 0; after the TC center passes, the sea level returns to its original value by the downward vertical velocity behind the TC center. The horizontal scale of the pressure core of the model TC is L ∼ 200 km and the pressure drop is ∼10 hPa (S1 Fig in S1 Text), the latter corresponding to a sea level rise of Δη = Δp/(gρ_o) ≈ 0.1 m. Therefore, w ∼ ΔηC_tc/L ∼ 0.4 × 10⁻⁵ m/s, consistent with Fig 4.

The vertical profile of this vertical velocity is linear: Ψ₀(z) = z/D + 1, where Ψ₀(z = 0) = 1. The magnitude of the vertical velocity is, therefore, w₀ at the surface, which is . The horizontal velocity (not shown) is vertically uniform (because ψ₀(z) = 1 = const.) and .

It is concluded that at the equilibrium, the pressure anomaly of the TC does not induce any significant disturbance to the ocean, except that sea level rises to cancel out the atmospheric pressure anomaly (η = η₀ = π₀/g = −π_a/g) and except for the associated vertical velocity to supply for the sea level rise and fall on the front and rear sides of the TC center. This state is often called “isostasy”. This property always holds for TCs because they are not nearly as fast as barotropic gravity waves and their core size is much smaller than the barotropic deformation radius.

3.2.2 Response to curl τ.

The barotropic response to a delta-function curl τ is described by [7]. Such a response is often called the “unit impulse response” or the “Green’s function”. Here, we extend his/her results to a more realistic wind-curl distribution.

Fig 5 shows the w₀ field of the barotropic response to the wind stress of the model TC. As can be inferred from the results of [7], the w₀ field is very nearly circular because curl τ is circular. Anisotropy results only from the anisotropic Laplacian (Eq. (S22) in S1 Text) but this anisotropy is negligible for the barotropic mode because .

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Fig 5. The distribution of w₀ from the mode-0 RGM forced by winds at the end of the integration; zonal and meridional sections across the center of the TC.

The orange curve is the analytic impulse response to a delta-function forcing at the center of the TC (Green’s function; Eq. S25 in S1 Text); its amplitude is adjusted so that the peak visually agrees with that of w₀, which is fine for a qualitative comparison like this. See Section S1.3.6 in S1 Text for a discussion about the amplitude.

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Since curl τ is concentrated near the center of TC (S1 Fig in S1 Text), one would suspect that the impulse response would be an excellent solution. It is therefore interesting that the w₀ solution (blue curves) spreads much less than the impulse response (orange curves). Fig 6 compares the impulse response with the actual distribution of w₀. This time w₀ has been temporary averaged to suppress gravity waves. Also plotted is a numerical solution to the “w-equation” (1) forced by the curl of the same wind stress as forces the RGM. (The numerical method is described in Section S1.3.5 in S1 Text). This solution is plotted with the thick dashed green curve, which agrees well with the RGM solution (blue curve). In contrast, the orange curve is the numerical solution to the w-equation forced by a delta-function-like curl: curl τ = const. within a 20 × 20 km square centered at the center of the TC and curl τ = 0 outside. The solution is then scaled in such a way that its peak visually agrees with the peak of w₀.

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Fig 6. Distribution of w₀ along y = 3000 km from the wind-driven RGM solution.

The solid blue curve is the same as that in the top left panel of Fig 5 except that this version is a temporal average over the last three days to suppress gravity waves. The orange curve is different: this time, it is a numerical solution to the “w-equation” (1) forced by a delta-function-like wind curl; its amplitude is adjusted so that the peak visually agrees with that of w₀. The thick dashed green curve is the numerical solution to the w-equation forced by the curl of the same wind stress as forces the RGM.

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Solutions to the w-equation can be viewed as a steady solution to the virtual “heat-conduction equation” (S29) in S1 Text, where positive wind curl is analogous to “heat source” and −w is analogous to “temperature”. The “heat” provided by the forcing is laterally diffused by the Laplacian term and gradually lost by the “radiative cooling” −α² ⋅ (−w). Since “cooling” is weak, that is, the radius of deformation (α⁻¹) is large, the anomaly spreads broadly.

The difference between the realistic w₀ response and the impulse response must therefore be attributable to the negative wind curl, “heat sink”, outside the TC core (S1 Fig in S1 Text). It is therefore the negative wind curl outside the TC core which limits the horizontal extent of w. Although weak in magnitude, the negative curl is effective (in absorbing “heat”) because it covers an area much larger than the area of positive curl. A region of negative curl is an integral part of a TC because its wind decays (or must decay) fast outside the TC core. See the discussion of S1 Fig in S1 Text.

As discussed for the p_a response in the previous subsection, the pressure, and equivalently sea-level, response is the zonal integration of w (Fig 7). The pressure starts to drop ∼400 km before the TC center and reaches its negative peak ∼400 km behind the TC center, corresponding to the zonal scale of w₀ (Fig 7). Since the speed of this TC is 8 m/s, these corresponds to ∼0.7 d before and after the passage of the TC center in the stationary system. The lateral extent is the same as that of w₀, that is, Δy ∼ ±400 km. In the wake of the TC, the low pressure stays at its peak value. Geostrophic circulation is left behind; (u, v) ≈ g(−η_y, η_x)/f is a very good approximation (not shown). The velocity is vertically uniform (because ψ₀(z) = 1) and its amplitude is less than 0.75 cm/s (not shown). The horizontal scale of the pressure anomaly and the associated velocity anomaly are therefore ultimately determined by the radial structure of the wind curl of the TC.

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Fig 7. The distribution of η₀ from the mode-0 RGM at the end of the integration; zonal and meridional sections across the center of the TC.

Since , of sea level anomaly corresponds to 1 hPa of pressure anomaly (note that ψ₀(0) = 1). The arrows indicate the horizontal velocity (in the stationary coordinates).

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3.3.1 Mode 1: w.

Fig 8 plots −w₁ from the n = 1 RGM and from the numerical evaluation of the analytic solution (Section S1.3.4 in S1 Text) to (1). The unfortunate consequence of the sign convention (ψ(0) > 0) and ψ’s relation with Ψ (Eqs. S6) in S1 Text is that Ψ_n(z) < 0 near the surface for all n ≥ 1; for this reason, w_n < 0 means upwelling near the surface.

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Fig 8.

Vertical velocity −w₁ of RGM mode 1 at the end of the integration (A) (B) (C) and the numerically evaluated analytic solution (D) (E) (F). See Eq. (S28) in S1 Text for the latter solution. The zonal section goes through the center of the TC (y = 3000 km) whereas the meridional section is x = 2850 km, where the peak of w₁ is. The impulse response (Green’s function, orange curves) is shifted westward by 30 km and its amplitude is adjusted, both in such a way that the curve visually agrees with the w₁ curve.

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As is well known, a typical TC leaves a train of near inertial oscillation behind, which is because C_tc > c₁ for a typical TC as explained in the following. The orange curve is a plot of the Green’s function centered at of the w-equation: (3) where J₀ is the Bessel function of the first kind of order 0 and (4) [7, and see Eq. (S26) in S1 Text of the present paper]. The actual solution is a convolution integral between the Green’s function and the wind-stress curl (Eq. S28) in S1 Text, but since the strong positive wind-stress curl is concentrated in a circle of a radius ∼100 km (S1 Fig in S1 Text) the Green’s function is a fairly good representation of the actual solution.

The meridional extent of the solution scales as α⁻¹ (= c/f). The wavelength of J₀(r*) is very nearly 2π near r* = 0 (not shown) and therefore the zonal wavelength at y* = 0 near x* = 0 is (5) which slowly increases as c_n decreases for higher modes. For mode 1, where c₁ ≈ 2.55 m/s, λ₁ ≈ 655 km. But, because usually for all baroclinic modes, and the structure of the solution depends only weakly on c in the x direction and the zonal wavelength is mainly determined by the speed of the TC. In the stationary coordinate system, because this wave train moves eastward at the speed of C_tc, the angular frequency of the oscillation is ω = 2πC_tc/λ ≈ f and ω > f, that is, this oscillation is near-inertial.

An obvious difference between the Green’s function and the actual solution (Fig 8A) is that the former extends westward indefinitely because it assumes a steady state in the moving coordinates. The system can propagate disturbance no faster than c and this explains the nearly perfect circle centered at (x, y) = (1000 km, 3000 km) (Fig 8B), where the TC initially was. Also, the meridional extent of w₁ at its peak is much narrower in the Green’s function (the orange curve in Fig 8C). This is again because the signal from the forcing, which is a delta function for the impulse response, cannot propagate faster than c. Moreover, to obtain a good match with the actual solution, the Green’s function has to be shifted westward by 30 km (the orange curve in Fig 8A).

These discrepancies indicate that the position of the peak in w₁ behind the TC and its meridional structure need to be explained by the convolution integral between the actual distribution of curl τ and G. Fig 8D to 8F show a numerical evaluation of the convolution integral, which we sometimes call the “analytic” solution for convenience. This solution is quantitatively very similar to the RGM solution. Fig 9 compares the two solutions in detail. The match is nearly perfect in the zonal section (Fig 9B) except that the first peak is somewhat larger in the “analytic” solution. The wavelength of the Green’s function is slightly longer than that of the actual solution (Fig 8A); this difference also comes from the difference between the Green’s function itself and the convolution. Since the actual solution is a westward integration of wind-stress curl times the Green’s function (Eq. S28) in S1 Text, the peak response in w_n is a westward accumulation of forcing curl τ and the peak comes west of the TC center, whereas the peak of the Green’s function itself is located right at the TC center. This is the reason for the 30km westward shift applied to the orange curve in Fig 8A.

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Fig 9.

Vertical velocity −w₁ of RGM mode 1 (a) at x = 2290 km, where the 9th peak is, (b) across the center of TC, y = 3000 km, and (c) at x = 4850 km, where the 1th peak is. The solid blue curve is the numerical solution to the RGM at the end of integration, and the thicker dashed green curve is the numerically evaluated analytic solution to the w equation (Eq 1; see Section S1.3.4 in S1 Text).

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The meridional structure at the first peak (Fig 9C) also agrees well between the two solutions. The Green’s function shows a clear causal relation (Eq. (S26) in S1 Text): No response shows up outside the causal triangle x* < y* < −x* with x* < 0. In dimensional variables, (6) from (4) because and α = f/c. The triangle extends from westward, northward, and southward. This triangular shape is a natural consequence of the condition that C_tc > c_n [7] but we are not aware whether it has been observed or not. If there are no observational reports, it may be because the near surface response is dominated by higher vertical modes (see Section 3.4 below), for which the meridional spread is narrower (see Section 3.3.3 below).

This is the bound where gravity waves can reach from the delta-function forcing. This is the reason why G has a sharp edge at the triangle. The actual response is smoother because the forcing is smooth but the amplitude of the response rapidly decays outside the triangle which starts from the TC center (Fig 8A and 8C) because the forcing (wind curl) decays rapidly away from the TC core.

The meridional spreading due to gravity waves, as represented by the width of the orange curve in Fig 8C, is narrower than the meridional scale of the wind curl itself and therefore the overall meridional width at this point (150 km behind the TC center) is determined mainly by the meridional scale of the wind curl.

Further west (Fig 9A), the upwelling or downwelling region spreads meridionally by gravity waves but the total response is a superposition of gravity waves generated at various places. (This superposition is what the convolution integral does.) The net result is a wavy pattern that includes small scale features near the edge (Fig 9A). It is interesting that at the poleward edges the “analytic” solution has small but nonzero amplitudes beyond the latitudes where the numerical RGM response vanishes. This may be because the numerical convolution integral must be carried out at a higher horizontal resolution to cancel out these tiny wiggles or because small-scale gravity waves are somewhat slower in the numerical RGM.

3.3.2 Mode 1: Other fields.

Fig 10 plots η₁ψ₁(0). We include the factor ψ₁(0) to indicate the contribution of the mode to the actual total sea level; see (S7) in S1 Text. As stated for the barotropic mode, w_n = −C_tc∂_xη_n at the equilibrium state on the moving coordinates and therefore the η_n field is proportional to a westward integration of w_n. For this reason, the peaks of η₁ is a quarter phase behind those of w₁. The location of the first minimum pressure (sea level) is therefore determined by both the horizontal size of the strong-curl core of the TC and the natural wavelength (2πf/C_tc) of the near-inertial oscillation. It is located 300 km behind the TC center, which corresponds to about 10 hr behind in the stationary coordinate system. Interestingly the response of pressure is not only a near-inertial oscillation but also there is a net pressure drop. The size of the average drop is less than 1/10 of the barotropic mode (Fig 7).

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Fig 10. Same as Fig 8 except that this one plots η₁ψ₁(0).

Note that the meridional section is still where w₁ is at its peak, not η₁.

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Since pressure (π), sea level (η), and buoyancy (b) are tied by π_n = gη_n = −Db_n for each mode (Eqs. S8e and S8g) in S1 Text, a negative pressure anomaly simultaneously means a negative sea level anomaly and a positive density anomaly near the surface (because ψ_n(0) > 0 according to our sign convention). Therefore, pressure anomaly near the surface is negative owing to the lowered sea level despite the positive density anomaly. The density response is shown later for the total 4-d field.

It is not easy to show the velocity field in arrows on the map (10) as it is dominated by eddy-like wave pattern as shown in [3]. Its magnitude is less than 0.8 cm/s (not shown).

3.3.3 Mode 10 and higher.

As the mode number increases (that is, as c_n decreases), the triangle of causality (6) shrinks in the meridional direction because c_n/C_tc decreases. For mode 10, for example, c₁₀ ≃ 0.25 m/s and c₁₀/C_tc ≃ 0.03, which means that even after 5000 km of TC passage the meridional spread of the signal is just ±160 km (Fig 11). As a result, the response to the forcing is dominated by the westward integration of wind curl with the Green’s function as in (S28) in S1 Text. That is why the meridional structure of w₁₀ near the TC center (Fig 12C) is similar to the meridional structure of wind curl, which has two peaks at Δy ≈ ±20 km (S1 Fig in S1 Text).

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Fig 11. Vertical velocity −w₁₀ and sea level η₁₀ψ₁₀(0) of RGM mode 10 at the end of the integration.

The zonal section goes through the center of the TC (y = 3000 km) whereas the meridional section is 170 km to the west, where the peak of w₁₀ is.

https://doi.org/10.1371/journal.pclm.0000376.g011

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Fig 12.

Vertical velocity −w₁₀ of RGM mode 10 (A) at x = 2050 km, where the 9th peak is, (B) across the center of TC, y = 3000 km, and (C) at x = 4830 km, where the 1th peak is. The solid blue curve is the numerical solution to the RGM at the end of integration, and the thicker dashed green curve is the numerically evaluated analytic solution to the w equation (Eq 1; see Section S1.3.4 in S1 Text).

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According to (5), the zonal wavelength near the TC center is approximately 690 km, somewhat longer than 655 km for mode 1. These wavelengths are not inconsistent with the distance between the 1st and 9th peaks in |w| quoted in the captions to Figs 9 and 12, according to which the wavelengths are 695km for mode 10 and 640km for mode 1.

The pressure response again includes a mean drop and is much smaller (Fig 11) than that of mode 1 (Fig 10). The velocity field again takes a form of series of eddies and is smaller than 1 cm/s.

Further reduction in c_n brings about even smaller changes to the response of the mode to wind curl. The meridional spread of signal becomes even narrower and as a result, so does the causal triangle, and the westward convolution integration dominates even more. In other words, the response of the mode converges as c_n becomes smaller. (If we looked at x → −∞, the signal always spreads meridionally for all modes, but we are interested in a limited range of x.)

Our numerical RGM solution agrees with the convolution integral at mode 20 (not shown) as well as it does at mode 10 (Fig 12) even though with c₂₀ ≈ 0.13 m/s, the deformation radius c₂₀/f ≈ 1.9 km is hardly resolved by our meridional grid spacing of 2 km. This result must be because the meridional spread of the solution (as embodied by the Green’s function) is negligible and the numerical grid does not need more meridional resolution than for lower modes.

3.4.1 w field.

Fig 13 shows the wind-forced 3-d solution at the end of integration. (The p_a-force w is negligibly small as discussed later in Section 3.6.1 and so is ignored here.) At about 150 km behind the TC center, there is a vertical column of strong upwelling, which grows linearly from 0 at the bottom to ∼4 × 10⁻⁴ m/s at z ≈ −250 m and then decays to 0 at the surface (Fig 13G). This profile is similar to that found in the JCOPE model (Fig 2) except that the depth of the peak appeared more like 700m in the latter. This vertical coherence in the idealized model is because all baroclinic modes have the first peak in w_n approximately here. The vertical structure is a result of constructive interference in the upper ocean and destructive interference at depth between the vertical structure functions Ψ_n(z) of the modes.

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Fig 13. Vertical sections of w(x, y, z) forced by winds at the end of integration.

The units for the contouring are 10⁻⁴ m/s. Arrows indicate velocity vectors (v, w) in each meridional section; velocity is omitted for the zonal section to avoid too much cluttering. The velocity components, v and w, are scaled in such a way that the arrows are parallel (tangent) to local stream lines, each pointing in the direction of the motion of the local water parcel. The zonal section (A) is through the center of the TC; the meridional sections are where the 9th peak is located for mode 10 (B) and for mode 1 (C) and where the 1st peak is for mode 1 (D); and the vertical profiles (E)(F)(G) are taken at y = 3000 km for these three sections. The position of the 1st peak for mode 10 is omitted because it is quite close to that for mode 1.

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Further behind, the phase lines tilt backwards. The zonal wavelength becomes longer as c_n decreases (Eq 5) but the rate of this change slows down as n increases (and λ_n approaches a constant value). For this reason, the vertical alignment of peaks in w_n (Figs 8 and 11) is gradually lost between the lowest modes and the higher modes whereas the coherence between the higher modes mostly remains, and the shallowest peak in w remains at ∼300 m, which is created by the sum of the higher modes where as the relation of the other, deeper peak(s), which are created by lower modes, to the shallowest peak, keeps shifting (Fig 13E and 13F).

The meridional evolution of the wave train is also explained as a superposition of modes. The meridional spread is obviously faster for lower modes (Eq 6; Figs 8 and 11), and as a result, the outer edges are dominated by a 1st-mode-like structure and variability is found only at mid depths away from the TC track (Fig 13B and 13C).

This 3-d structure should also be explainable as a behavior of 3-d inertio-gravity waves because after the passage of the TC, the w disturbance left behind should follow the dispersion relation of internal gravity waves. Although the 3-d disturbance field is a superposition of waves with a wide range of vertical wave numbers and does not render itself to a simple WKB interpretation, at least the propagation of the phase is consistent with the interpretation that this is a kind of lee wave response: We have plotted x-t and z-t Hovmöller diagrams at a fixed x behind the center of TC (not shown) and found that the phase propagates upward and forward (positive x), consistent with the regular lee wave [e.g., 18].

This indicates a downward and forward energy propagation [e.g., 15]. Using the standard dispersion relation [e.g., 15] of internal gravity waves and for the 1st baroclinic mode, which has a vertical wavenumber of m = π/D (Eqs. S20) in S1 Text, a zonal wavenumber of (Eq 5), and a meridional wavenumber of l = 2π/800 km from a rough estimate of the half-amplitude meridional width of 400 km (Fig 12), we calculate the group velocities (The horizontal group velocities can be calculated from the standard dispersion relation ω² = f² + c²(k² + l²) of the inertio-gravity wave (Poincaré wave) of the single-mode RGM because c₁ = ND/π = N/m for mode 1. This is because the 3-d internal-wave dispersion relation reduces to that of the Poincaré wave in the limit that k² + l² ≪ m².) Likewise, for mode 10 as another example, Morozov & Velarde [19], analyzing observed near-inertial waves generated by a typhoon near Japan, reported that the vertical propagation speed is 1–10 m/hr (2.8 × 10⁻⁴–2.8 × 10⁻³ m/s). Because the wave pattern is a superposition of vertical modes with a wide range of wavenumbers, the range of the propagation speed is large. One caveat is that our linear model misses the potential trapping of small-scale near-inertial waves owing to the relative vorticity [e.g., 20].

In the coordinate system moving with the TC, energy propagates backwards away from the TC, which corresponds to the typical lee-wave situation, where winds blow over a mountain. When the wave maker is moving, however, the energy propagates toward the wave maker [e.g., 18] in the stationary coordinate system. One interesting difference from typical lee waves is that the TC generates anomaly from the surface to the bottom at once as a column, which is the reason why disturbance shows up near the bottom relatively quickly despite the slow vertical group speed.

It is interesting that the meridional profile (Fig 14B) of w at a shallow depth retains the double peak structure of higher modes (Fig 12C).

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Fig 14. Zonal (y = 3000 m) and meridional (x = 4850 m) profiles of w (blue) at z = 300 m, where w is maximum (see Fig 13G).

The latter section is where the first peak in w is. The orange curves are from the analytic solution (S45) in S1 Text of the uniform-density ocean. This point is discussed later.

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3.6.1 Sea surface response.

Since we include only mode 0 for the p_a-forced solution, the 3-d structure of the p_a-forced w is the 2-d structure of Fig 4 times the vertical structure functionΨ₀(z) = z/D + 1 of mode 0. This w field is more than 10 times smaller than that driven by winds (Figs 8 and 11). We actually tried adding this w field to the wind-driven w above, but the result was visually indistinguishable from that of the wind-forcing-alone solution except very near the sea surface.

Even smaller is the barotropic, wind-driven w. Although too weak to be visible in Fig 13, w < 0 at the surface for the wind response. This feature is solely due to the barotropic mode (Fig 5) as Ψ₀(0) = 1 and Ψ_n>0(0) = 0.

In contrast, the sea level anomaly is utterly dominated by p_a forcing (Fig 3). The associated surface pressure anomaly, however, is almost totally canceled by p_a itself (Fig 3) and the sea-level pressure anomaly is dominated by the barotropic response to winds (Fig 17) followed by the baroclinic response.

3.7.1 Stationary wind curl.

The theoretical steady response of a flat-bottom, uniform-density linear ocean to a stationary, steady wind is well known (Section S1.5.3 in S1 Text). Eq (7) reduces to the familiar Ekman pumping w_E = curl τ/(ρ_of). Since below the mixed layer, geostrophy gives w_z = −(u_x + v_y) = 0, a vertically uniform column of vertical velocity equal to the Ekman pumping is expected [Section S1.5.3 in S1 Text; 21, see his Fig. 4.13.3]. To reconcile with the bottom boundary condition that w|_z=−D = 0, a bottom boundary is necessary; otherwise, this state cannot exist.

3.7.2 Equilibrium response.

If we relax the requirement that ∂_t = 0 for all variables and assume only that ∂_tw is constant in time, (7) reduces to the familiar Ekman pumping w_E = curl τ/(ρ_of) again and we obtain an equilibrium solution Eq. (S42) in S1 Text, which satisfies w|_z=−D = 0. To reproduce it here, (8a) (8b) That is, w has a linear profile up to the bottom of the Ekman layer.

The problem with this solution is that the sea level keeps dropping because of the Ekman divergence. As seen below, the response to a moving TC is analogous to this state except that the sea level does not keep dropping there because the TC is moving.

3.7.3 Moving TC.

Here we discuss only the results; the derivation is detailed in Section S1.5.4 in S1 Text. When the TC is moving at a steady speed of C_tc, the vertical profile of w is still linear as in (8b), and (7) can be written as (9) in the moving coordinates. The solution can be written as a zonal convolution integral between curl τ and the Green’s function (S45) in S1 Text. This integral is again easily evaluated numerically on a grid as discussed (Eq. S1.3.4) in S1 Text for the solution shown in Fig 9. This solution agrees fairly well (Fig 17) with the actual w at z = 200–300 m, where w takes its peak value (Fig 13G).

In the uniform-density solution, the oscillation continues at a constant amplitude and its wavelength is C_tc/|f| (Eq. S45) in S1 Text, which is the wavelength in the limit that , a good approximation for a typical TC translation speed (see Eq 5). Indeed, in that limit, the “w-equation” (1) for each mode n reduces to which indicates that for all n ≥ 1. As we mentioned earlier, our sign convention means that −w_n is proportional to the mode’s contribution to the 3-d w and the factor is the nondimensional weight for each mode. In this limit, the peaks of all baroclinic modes coincide with that of the delayed Ekman pumping (9) and the resultant perfect vertical alignment would lead to the columnar w.

These observations suggest that the uniform-density solution can be viewed as a limit that c_n → 0 for all baroclinic modes (n ≥ 1). That interpretation is supported by the result shown in Section S1.2.5 in S1 Text that in the limit of weak stratification, the baroclinic c’s all approach zero with the eigenfunctions changing little, whereas the barotropic mode changes little.

In the actual solution, the amplitude slowly decays to the west because the vertical modes gradually lose vertical alignment (Section 3.4.1). In the meridional profile (Fig 14B), the double-peak structure, reflective of the structure of the wind curl (Fig 11), is more pronounced in the delayed Ekman pumping than in the actual w. This must be because with a finite c_n, the impacts of curl are spread meridionally by inertio-gravity waves, whereas for the uniform-density ocean, solution takes the form of a pure zonal integration of wind curl (Eq. S45) in S1 Text.

As expected, the sea level response of this uniform-density solution is almost identical to that of the barotropic mode (Section S1.5.4 in S1 Text).

4.1.1 Barotropic response.

The barotropic mode responds to the TC’s low pressure (p_a) with an almost matching sea level rise, canceling the low pressure and generating little horizontal flow (Fig 3). This “isostatic” balance holds because and , where is the barotropic gravity-wave speed, L is the horizontal characteristic scale of p_a, and α⁻¹ ≡ c₀/f is the barotropic Rossby radius. Both conditions are usually satisfied for TCs. The only significant response is the upwelling that feeds the sea-level rise in front of the TC and the matching downwelling that restores the sea level behind the TC (Fig 4). It is (Fig 4) and linearly decreases to zero at the bottom. This is not insignificant but it is an order of magnitude smaller than the baroclinic w to be discussed below.

The barotropic response to the TC’s wind stress is a permanent sea level drop due to the (barotropic component of) Ekman divergence behind the TC (Fig 7). The resultant pressure anomaly is roughly ∼2 hPa. The vertical velocity associated with this motion is a very small downwelling which is at the surface and linearly decreases to zero at the bottom (Fig 5). The horizontal scale of the pressure drop, ∼400 km, is significantly limited by the negative curl surrounding the positive curl of the TC core (Fig 6). This is because even though the amplitude of the negative curl is weak, its area is much broader than that of the positive core.

4.1.2 Baroclinic response.

The baroclinic response to p_a is negligible because this forcing term enters the primitive equation as a vertically-uniform body force (Eq. S3) in S1 Text, whose contribution to each mode is proportional to the vertical integration of the p-mode functions, ψ_n(z), and the vertical integration is negligibly small for all baroclinic modes.

The baroclinic response to winds takes a form of the well-known train of waves (Figs 8 and 11). Its streamwise wavelength is approximately , and the angular frequency is . As the mode number n is increased, the zonal wavelength gradually increases and approaches the constant value 2πC_tc/f, and the frequency gradually decreases and approaches the constant value f. Both are, however, already close to the final value because .

The first peak of upwelling comes about a quarter wavelength behind the center of the TC (Figs 8 and 11) and a regular oscillation in w follows. The net impact of the wave train is a permanent pressure drop in the upper ocean, lowered sea level, and cooling near the surface (Figs 10 and 11).

The lateral scale of the first upwelling peak is almost that of the TC’s core of positive curl τ (Figs 8C and 9C) because the lateral scale of the meridional propagation (Eq 6) is smaller than that of the TC’s core at this point. The subsequent peaks in the wake gradually spread as a packet of inertio-gravity waves; this spread is naturally larger in lower modes (Fig 9A) and smaller in higher modes (Fig 12A).

4.1.3 4-d structure.

The most striking feature of the superposed w field (Fig 13) is the column of upwelling that linearly grows from zero at the bottom to its maximum near the bottom of the mixed layer (Fig 13G). This structure is a result of the vertical coherence of the modes: all baroclinic modes have the first peak in w_n approximately here. This vertical coherence is gradually lost because the zonal wavelength lengthens for higher modes. The lateral spread of the wave train also reflects the speed difference between modes (Eq 6; Figs 8 and 11).

Viewed as a collection of near-inertial waves, this pattern may be interpreted as a “lee wave” response, except that the wind curl of TC initially generates the disturbance throughout the water column, not just within the mixed layer. For this reason, the near-inertial waves appear near the bottom quickly without waiting for the wave packets to propagate downward at the very slow vertical group velocity.

As expected, upwelling wins on average and the wave train leaves a region of colder water. The density anomaly increases from zero at the bottom to its maximum value at the bottom of the mixed layer (Fig 15). The lateral convergence of water that feeds the upwelling creates this density structure.

The pressure anomaly is dominated by the barotropic response to wind curl. The pressure drop starts ∼400 km before the TC center and completes ∼400 km after the center (Fig 16). The lateral spread is of a comparable scale (Fig 16) because the w response is approximately circular (Fig 5).

The baroclinic modes modifies this pressure field with their near-inertial oscillation. Since this oscillation starts with negative and positive pressure anomalies in the upper and lower oceans (Fig 16E), it reinforces and weakens the barotropic pressure anomaly in the upper and lower oceans, respectively (Fig 17). The first negative peak in pressure anomaly at the bottom is delayed (Figs 16A and 17B) as a result.

4.1.4 Uniform-density ocean.

The same moving wind stress curl drives a train of inertial oscillation in the linearized primitive equations without stratification (N = 0). The streamwise structure of this w field is a simple inertial oscillation very similar to that of the numerical solution (Fig 14A) and its vertical structure is the same as that of the first upwelling column of the numerical solution (Fig 13), that is, a linear profile from the bottom to the bottom of the mixed layer. This solution is a response to the “delayed Ekman pumping” (7) and can be regarded as a limit that c_n → 0 for n ≥ 1 and thus explains the solution with a stratification because for n ≥ 1.

4.2.1 Typhoon composites.

The composite vertical profile of w from the OGCM in Fig 2 has some similarity and some discrepancies with that of our idealized theoretical calculation in Fig 13G. The theoretical amplitude, which is the same as that of the delayed Ekman pumping according to our interpretation (Section 3.7), is 4 × 10⁻⁴ m/s and that of the composite (Fig 2) is about 1 × 10⁻³ m/s or somewhat larger.

The central pressure of this TC is about 985 hPa at the time of Fig 2A and it was about 960 hPa one day before when the TC center was at about 26°N. The ambient pressure, though hard to estimate visually, seems to be about 1015 hPa in Fig 2A. That is, ΔP ∼ 30–55 hPa, which is 1.5–2.5 times the value we have used for the theoretical calculations. Near the center of the TC, where the centrifugal force dominates [e.g., 22, and references therein], the squared wind speed, and hence the wind stress, is proportional to ΔP (Section S1.4 in S1 Text) and therefore the magnitude of w is roughly proportional to ΔP. The theoretical amplitude is, hence, not inconsistent although more detailed comparison would be necessary to give a more reliable answer.

The peak depth is ∼1000 m for the composite (Fig 2B). The mixed layer cannot be this deep and, if our interpretation of the depth profile is correct, this discrepancy must be because of noise, including tides and other flows, which are not directly driven by the TC.

At this point, however, we do not attempt further detailed comparisons because we do not know how representative the composite Fig 2 is. We just show the one that looked the most clean among the several composites we looked at. Although strong vertical velocity reaching near the bottom was a common feature, the vertical profiles varied (not shown).

JCOPE-T includes tidal forcing generating barotropic tides, and large-scale internal tides are also abundant in the model [23, 24]. This may be one of the reasons why the composites differ so much between different typhoons (not shown).

To find common features, we trialed composites of low-pass-filtered data. Specifically, we applied a Hann filter with a half window size of 24 hours at each spatial point before calculating the temporal anomaly. That is, where the period of the temporal average is ±4 d around the TC’s time. The composite now showed a much smoother and more systematic vertical velocity field (not shown). Since the timescale of smoothing much reduces the near-inertial oscillation, the vertical profile of this smoothed field is no longer that of the first upwelling peak. Also, the direction of the zonal and meridional sections are different from the angle of the TC track and therefore the temporal average includes variability somewhat outside the variability directly behind the TC.

Also, many typhoons rapidly change their courses or their speeds or both during their traversal south of Japan. (One can confirm this by looking at the “best track” dataset described in Section S1.1.1. in S1 Text) Composites like Fig 2 may not capture the core of the upwelling column.

The best approach would therefore be 1) use another high-resolution model that does not include tides, 2) choose TCs that traveled in a relatively straight line, 3) discard the portion of the track where the TC’s speed changed drastically, and 4) set the vertical section along the track, not in the meridional or zonal direction. This would be an interesting future project.

4.2.2 TC speed.

Our analysis has been benefited from the parameter range that for all n ≥ 1. (Even though c₁ ≪ C_tc does not hold, does.) When C_tc is faster, not much changes (as long as ). The dependency of the frequency and wavelength of the near-inertial oscillation on c_n will be even weaker (see Eq (5) and discussion below it) and the frequency will become even closer to f. Consequently the dispersion of the near-inertial waves behind the TC (Fig 13) will be weaker still. The triangle of causality (Eq (6), the wedge shape behind the TC in Fig 8) will become narrower. When C_tc is slower, the frequency and wavelength will become somewhat more sensitive to c_n (Eq (5)) and the inertial oscillation will be somewhat more dispersive. The triangle of causality will become wider.

An interesting question is what happens when C_tc ≈ c₁, although such a situation is rare [6]. Trying to see what happens, we ran our RGM with C_tc = c₁. The response was weak and noise was relatively large (not shown). This is understandable. As C_tc → c, the Green’s function for C_tc > c (Eq. S25 in S1 Text) becomes wigglier (its wavenumber becoming shorter and shorter) in the x direction and that for C_tc < c (Eq 3) becomes narrower. Finite-difference error would inevitably contribute. To solve this problem for a finite-difference model, one would need to introduce some form of spatial mixing.

On the other hand, this is not a resonance. When C_tc = c_n, the left-hand-side operator of the w-equation (1) becomes and allows for a finite solution with a simple Green’s function (not shown). Because there is no x dependency, the solution would retain the x structure of the forcing and integration happens only in the y direction.

One approach would therefore be to replace the numerical RGM with convolution integrals like (S28) in S1 Text for all baroclinic modes. To obtain other fields than w, one would use the convolution integrals of [12]. Even though the numerical RGM is much more versatile, being applicable to a much wider variety of situations (what if the TC’s track isn’t straight, for example?), this kind of semi-analytic approach would cover a wider range of C_tc for the simplest situation we have been looking at.

4.2.3 Limitations and potential extensions.

Obviously, our linear model lacks horizontal advection of momentum and density and therefore nonlinear impacts such as enhanced mixing [25] are missing. The potential trapping of small-scale near-inertial waves by relative vorticity [20, 26, e.g.] is also missing. Jaimes & Shay [27] reported upward energy propagation of near-inertial waves with wavelengths of ∼ 200 –300 m in their observation, whereas our results include only a large-wavelength near-inertial waves whose energy propagates downward. This feature could also be due to some subsurface nonlinear processes.

The large-scale (lower-mode) part of the inertial oscillation (Fig 13) would probably be affected less by advection because of its higher group speeds, but how nonlinearity affects the near-bottom anomaly field would be an interesting subject of future studies.

Another process missing from our model is the deepening of the mixed layer due to the increased turbulent mixing [e.g., 3, and references therein]. Assessing this impact on the basis of the present study’s results, we conjecture that the deepening can be simply handled by changing h_m. Since this is a vertical mixing, the overall weight of the water column is not affected and the impacts on the bottom pressure would be small.

We used a constant stratification for simplicity and in hindsight this was a good choice because a more realistic stratification would not bring about something significantly new. The relation between n and is no longer linear with a more realistic stratification, but the property that c_n decreases as n increases stays. Because of the different n–c_n relation, the phase lines behind the TC in Fig 13A would take somewhat different curvatures and vertical profiles of w in the wake further down (after the dispersion of the vertical modes) would be different. For example, The vertical profile of w away from the TC track in Fig 13B is similar to sin πz/D because that is the vertical profile of Ψ₁(z) for mode 1 with N = const. With a more realistic N(z) with a typical pycnocline, the vertical profile of mode 1 has its peak near the pycnocline [15]. Crucially, however, the vertical column of w just behind the TC would not change depending on N (Section 3.7).

Presence of lateral boundaries would not affect the equilibrium response to TC since the main oceanic response is the columns of near-inertial oscillation, which do not propagate far, behind the TC.

If the planetary beta is included, the anomalies behind the TC would slowly propagate as Rossby waves. In addition, if the TC moves in the meridional direction, the frequency of the near-inertial oscillation behind the TC should depend on latitude. It would be interesting to explore how these elements modify the solution we have obtained.

Finally, the flat bottom assumption was necessary for the modal-decomposition approach to work. The column of strong vertical velocity may induce strong currents on bottom slopes, which in turn may affect bottom pressure anomaly.

4.2.4 Bottom pressure.

In the introduction, we mentioned an observation in which the bottom pressure starts to decrease as a TC’s center approaches and reaches its minimum (Δp ∼ 2 hPa) 1–2 d later. Although our result shown in Fig 16 is not inconsistent with the observation, there is still too much uncertainty to draw any conclusion from this comparison.

We looked at the vertical velocity anomalies in JCOPE-T and recognized strong w anomaly extending upward from the Bonin Trench slope (not shown) next to the observational site. This feature may be an indication of enhanced local circulation, which may or may not be due to the TC and may or may not be affecting the bottom pressure anomaly.

The motivation of this comparison is two-fold: one is to validate and extend our simple linear model and the other is to provide the observationalists with information as to what signal one would expect from TCs. For both purposes, we would need to look at more observational data and to analyze OGCMs in more details.