The influence of spatial variations of the oceanic mixed layer depth (OMLD) on tropical cyclones (TCs) is investigated using a coupled atmosphere–ocean model. The model consists of a version of the Naval Research Laboratory limited area weather prediction model coupled to a simple 2½-layer ocean model. Interactions between the TC and the ocean are represented by wind-induced turbulent mixing in the upper ocean and latent and sensible heat fluxes across the air–sea interface.
Four numerical experiments are conducted with different spatial variations of the unperturbed OMLD representing idealizations of broad-scale patterns observed in the North Atlantic and North Pacific Oceans during the tropical cyclone season. In each, the coupled model is integrated for 96 h with an atmospheric vortex initially of tropical storm intensity embedded in an easterly mean flow of 5 m s−1 and located over an oceanic mixed layer that is locally 40 m deep. The numerical solutions reveal that the rate of intensification and final intensity of the TC are sensitive to the initial OMLD distribution, but that the tracks and the gross features of the wind and pressure patterns of the disturbances are not.
In every experiment, the sea surface temperature exhibits a maximum induced cooling to the right of the path of the disturbance, as found in previous studies, with magnitudes ranging from 1.6° to 4.1°C, depending on the initial distribution of the mixed layer depth. Consistent with earlier studies, storm-induced near-inertial oscillations of the mixed layer current are found in the wake of the storm.
In addition, numerical experiments are conducted to examine sensitivity of a coupled-model simulation to variations of horizontal resolution. Results indicate that the intensity and track of tropical cyclones are quantitatively sensitive to such changes.
Tropical cyclones (TCs) develop only over warm oceans and rely for their intensification and maintenance on latent heat release in cumulus convection that fuels the system. With a change of sea surface temperature (SST), both latent and sensible heat transfer from the ocean to the atmosphere, which provide the moisture supply and instability for the development of tropical cyclones, can be greatly affected.
Storm-induced SST variations depend primarily on upper-ocean dynamics. While TC-induced upwelling tends to reduce the oceanic mixed layer depth (OMLD), wind-induced turbulent mixing entrains cooler ocean water from the thermocline into the upper-oceanic mixed layer and tends to increase the OMLD, resulting in the cooling of the sea surface and a concomitant decrease in the intensity of the TC. This negative feedback mechanism has been established by a number of studies (e.g., Chang and Anthes 1979; Sutyrin and Khain 1984; Bender et al. 1993).
With the exception of the work of Hong et al. (2000), who investigated the response of Hurricane Opal (1995) to a transient mesoscale warm eddy in the Gulf of Mexico using a coupled three-dimensional model, the initial oceanic mixed layer depth in previous studies of TC–ocean interactions (e.g., Chang and Anthes 1979; Sutyrin and Khain 1984; Gallacher et al. 1989; Schade 1993) was specified as uniform over the entire model domain. In both the Pacific and Atlantic Oceans, however, the distributions of the OMLD exhibit considerable nonuniformity over distances of one thousand to a few thousand kilometers. During a typical month of the TC season, for example, the following features of the OMLD distribution are observed. At 25°N in the Atlantic, the OMLD is fairly uniform from 75° to 45°W (Fig. 1a), but at 40°W (Fig. 1d) it undulates in the north–south direction with maxima in the vicinity of the equator and 22°N and minima around 10° and 35°N. At 15°N in the eastern North Pacific, there is a monatonic decrease of the mixed layer depth from 160° to 120°W (Fig 1b). At 160°E in the western North Pacific, there is an even more pronounced decrease of the OMLD from the equator to 40°N (Fig. 1c). More generally, the monthly mean OMLD distributions show remarkably nonuniform features in both the Pacific and Atlantic (Wyrtki 1964; Bathen 1972; Levitus 1982), and the spatial variations can be expected to be even greater on individual days.
In this study, a simple 2½-layer primitive equation ocean model is coupled to a version of the Naval Research Laboratory (NRL) limited-area weather prediction model (Chang et al. 1989) and the governing equations are discretized on a common grid on a polar stereographic projection such that no interpolations are needed to evaluate the required fluxes between the atmosphere and the ocean. We investigate, through a series of numerical experiments with the coupled model, the differences in intensification and motion of an idealized TC traveling over an ocean with initially uniform and nonuniform oceanic mixed layers with variations on the order of those observed in the Pacific and Atlantic Oceans. The purpose of this paper is to investigate the influences of such variations on the intensification and motion of TCs with solutions of a first-order approximation, so that results of this study may serve as a benchmark for further investigations using more complex models with sophisticated physics when deemed as necessary.
The outline of the paper is as follows. The coupled model is described in section 2 and the experimental design in section 3. Baroclinic response of the ocean to a TC with fixed intensity is discussed in section 4. The mutual response of the TC and the ocean, with different mixed layer depth distributions, is presented in section 5, and concluding remarks are provided in section 6.
2. The coupled TC–ocean model
a. The atmospheric model
The atmospheric component of the coupled model is a version of the NRL limited-area weather prediction model described by Chang et al. (1989). It is a 3D, hydrostatic, finite-difference, primitive equation model with surface fluxes specified by bulk aerodynamic formulas and convection parameterized by Kuo’s (1974) cumulus parameterization scheme. There are 10 σ (=p/ps) layers in the vertical with equal intervals (Fig. 2), where p and ps are the atmospheric pressure and surface pressure, respectively. Arakawa’s staggered C-grid is used for the horizontal discretization. The model covers an area of 5000 km east–west by 4000 km north–south over the ocean with a grid resolution of 40 km. The southern boundary of the model domain is at the equator. Sponge layer conditions similar to those specified by Perkey and Kreitzberg (1976) are applied for damping the reflection of waves from the lateral boundaries. The model has been used to investigate the East Coast snowstorm of 10–12 February 1983 (Chang et al. 1989), the outflow layer of a TC (Shi et al. 1990), and the role of environmental asymmetries in Atlantic hurricane formation (Pfeffer and Challa 1992). Initial conditions for the atmosphere and the vortex are described in section 3. Documentation of the model is given in Madala et al. (1987). A version of the model in polar stereographic coordinates (Mao 1994) is used in this study.
b. The oceanic model
The oceanic component of the model (Fig. 2) is a 2½-layer primitive equation model similar to the one developed by Mao (1994). It consists of an upper mixed layer (ML), in which the fluid is fully mixed and the temperature is vertically homogeneous; a thermocline layer (TL), in which the temperature is taken to be constant during the integration; and an abyssal layer (AL), in which the fluid is assumed to be quiescent and unable to sustain a pressure gradient. Only baroclinic modes are retained in the system. The fast barotropic mode is excluded by using a rigid-lid approximation. The governing equations for the vertically averaged motion in the ML and TL are given in the appendix. The numerical values of the various parameters are listed in Table 1.
c. Atmosphere–ocean interactions
Interactions between the atmosphere and the ocean are specified through momentum, latent heat, and sensible heat fluxes across the air–sea interface. The mean values of surface pressure (Ps), eastward and northward wind components (u and υ, respectively), temperature (T), and specific humidity (q) at the lowest level of the atmospheric model are used to evaluate these fluxes. The latent and sensible heat fluxes from the ocean to the atmosphere, which depend on the SST, affect the energy supply to the storm. The momentum flux from the atmosphere to the ocean induces turbulent entrainment in the ocean from the thermocline layer to the mixed layer, thereby altering the SST.
Computational stability is ensured, unless specified otherwise, with the use of a 60-s time step in the atmospheric model and a 900-s time step in the ocean model. The much larger time step in the ocean model, in which the external gravity mode has been eliminated, is permitted because of the small phase speed (∼1 m s−1) of the internal gravity waves in the ocean. Numerical simulations are carried out up to 96 h in all the experiments.
3. Experimental design
The ocean model is tested first through an idealized experiment, in which a TC of fixed intensity is permitted to travel over the ocean at a constant speed for 96 h. Baroclinic response of the ocean to this external forcing is examined by comparing variations of the ML current, SST, and vertical velocity at the base of the ML with those reported in the previous studies.
Mutual response of the tropical cyclone and ocean to variations of the mixed layer depth are investigated through four numerical experiments using the coupled model. These experiments are summarized in Table 2. The first, E1, is a control run in which the coupled model is integrated with the initial mean oceanic mixed layer depth taken constant at 40 m (resembling that in Fig. 1a) and remaining at this depth during the simulation. Past studies have demonstrated that the interaction between a TC and the ocean results in a lowering of the SST, and that this cooling diminishes the intensification of the TC. The purpose of experiment E1 is to determine the magnitude of this effect in the present model. The results are then used as a benchmark for comparison with the remaining experiments in which the initial OMLD is taken to be spatially variable.
Here (xc, yc) is the geographical center of the model domain.
In all experiments, the initial SST is taken to be everywhere equal to 29°C, the initial wind to be easterly at 5 m s−1, and the initial location of the TC to be at x = 4000 km, y = 1600 km, where the OMLD is locally 40 m. The mean hurricane sounding of Sheets (1969) is used for the initialization of the atmospheric model and the initial vortex,
is similar to that prescribed by Sundqvist (1970) and Madala and Piacsek (1975), modified to include the vertical structure function S(σ). Here, υθ is the tangential wind velocity and υθmax is the maximum velocity in the vortex, located at radius ro = 300 km. The vertical structure function S(σ) has the value 1.0 for σ > 0.6 and decreases for σ < 0.6 to slightly negative values in the upper troposphere and lower stratosphere. While this function differs from the one used by Shi et al. (1990), it accomplishes the same purpose (viz., it permits an efficient development of the outflow at upper levels). In all of the experiments, perturbations of the initial mixed layer depth are initially zero and the ocean currents in the mixed layer and thermocline layer associated with the pressure gradient forces are initially in geostrophic balance given by
in the mixed layer and
in the thermocline layer. Here m is the map scale factor given in the appendix.
4. Baroclinic response of the ocean to TC
To evaluate the performance of the ocean model, we first confine our attention to the baroclinic response of the model ocean to idealized external forcing by a TC of fixed intensity traveling from east to west at 5 m s−1. The initial condition of the vortex is similar to that described in section 3, except that the vortex is centered at x = 4000 km, y = 2000 km and the maximum tangential velocity of the vortex is 20 m s−1 and located at radius ro = 200 km. Only winds at the lowest σ layer are used to drive the model ocean. The near sea surface air temperature is 23°C. The environmental mean flow is a 5 m s−1 easterly wind over the entire model domain. The initial ocean temperatures are prescribed at 29°C in the ML and 20°C in the TL. Variation of the initial OMLD is prescribed by Eq. (1). Initial ocean currents are in geostrophic balance and governed by Eqs. (5)–(8). Since the vortex is traveling 5 m s−1 westward, the wind field is shifted one grid to the left of the model domain every 10 time steps.
Figure 4 shows the results of the ocean model experiment at 96 h. The wind field is exhibited in Fig. 4a, in which the vortex traveled 1728 km from its initial location. The wind stress causes cyclonic momentum to be transferred from the storm to the mixed layer. As ML current increases, the Coriolis force and the nonlinear transport grow in magnitude, the former developing faster than the later and becoming the dominant term contributing to the momentum tendency by t = 6 h. During this time, the momentum exchange between the ML and TL through the shear stress sets the TL into motion. Subsequently, the pressure gradient force increases rapidly and, by 15 h, roughly balances the Coriolis force in both ocean layers. By 24 h, the maximum currents in the ML and TL reach 1.64 m s−1 and 0.1 m s−1 (not shown), respectively. These magnitudes are within the range of previous investigations (Chang and Anthes 1978; Price 1981; Sanford et al. 1987; Cooper and Thompson 1989). At 96 h, the maximum current has decreased to 0.92 m s−1 in the ML (Fig. 4b) and increased to 0.29 m s−1 in the TL. These changes are caused by the TC moving toward region of deeper (shallower) ML (TL).
Variation of the SST is demonstrated in Fig. 4c. The maximum cooling reaches −4.3°C at t = 96 h, which is located to the rear right of the vortex center about 160 km away from the track. The rear-rightward bias of the SST response was verified by both observations (Wright 1969; Pudov et al. 1979; Black 1983; Sanford et al. 1987; Shay et al. 1989) and previous numerical investigations (Chang and Anthes 1978; Price 1981; Khain and Ginis 1991; Bender et al. 1993).
Figure 4d shows the vertical velocity pattern at the base of the mixed layer at t = 96 h. There is strong upwelling at and around the vortex center. As the TC moves westward, it leaves behind alternating regions of upwelling and downwelling associated with inertia–gravity wave activity. The maximum upwelling and downwelling velocities, which are found in the regions of strongest divergence and convergence, are more intense on the right side of the TC path where the ML currents are faster. These phenomena are consistent with those found by Chang and Anthes (1978), Price (1981), Greatbatch (1983), and Shay et al. (1990).
5. Mutual response of the tropical cyclone and ocean to variations of mixed layer depth
The time variations of the maximum wind speed at 850 hPa and minimum surface pressure for each of the four experiments are shown in Figs. 5a and 5b, respectively. Significant differences begin to show up after about 48 h. We see here that the rate of intensification, as well as the maximum wind speed and minimum surface pressure attained, are greater in magnitude in E2 (LUP, in which the disturbance travels over an ocean with increasing mixed layer depth) than those in E1 (CTL, the control run in which the mean mixed layer depth is initially uniform). The opposite is true in E3, where the OMLD decreases in the direction the disturbance travels (LDN), thereby allowing greater cooling of the sea surface due to turbulent mixing of thermocline water into the mixed layer. The weakening that takes place after the disturbance reaches maximum intensity is also greatest in E3, since the disturbance reaches the region of shallowest depth and, therefore, lowest SST in this stage of its life cycle. In E4 (LUD, the mean OMLD increases first and then decreases in the direction the disturbance travels), the rate of intensification is more rapid than in the other three cases during the period from 44 to 71 h, while the disturbance travels toward the region of maximum OMLD, and the weakening is more rapid after this time, as the disturbance travels over a rapidly decreasing OMLD. Although the TC in E4 reaches its minimum Ps lower than that in E2 around 71–72 h, its corresponding maximum wind speed does not match up to that in E2 at the corresponding hours. This may be because the maximum wind speed in Fig. 5a is instant and it usually fluctuates more heavily than surface pressure does in a TC. On the other hand, the TC in E4 does maintain a period of maximum wind speed longer than that in E2, suggesting that the overall intensity of the TC is greater in E4 than in E2. This becomes clear after computing a mean kinetic energy index (MKEI), defined as
for TCs in all the experiments (Fig. 6), where Vmax(j) is an instant maximum wind speed of the TC and N is the length of TC’s life cycle.
In Figs. 7a and 7b, we see the track of the disturbance and the surface wind and pressure patterns, respectively, at the end of 96 h in experiment E1. Although the rates of intensification, maximum intensity, and subsequent rates of weakening of the disturbances in the various experiments are different from one another, the wind and pressure patterns in each of the other experiments (not shown) all resemble that in E1 and the tracks (not shown) are not significantly different from one another.
The variations of the maximum current speed induced in the mixed layer by the tropical disturbance as it develops and travels are shown in Fig. 8 for the different experiments. It is seen that, as the tropical cyclone increases in intensity, the mixed layer current increases more slowly when the disturbance moves toward a deeper mixed layer (E2 vs E1), and more rapidly when it moves toward a shallower mixed layer (E3). In experiment E4, the current slightly decreases from 24 to 48 h and then increases rapidly from 48 to 72 h, finally becoming as vigorous as that in E3. The velocity field in the mixed layer corresponding to the CTL experiment (E1) at 96 h is shown in Fig. 9. The flow pattern, with large velocity vectors and regions of convergence and divergence to the right of the storm path, is the same for all experiments (with different magnitudes) and is similar to the inertia–gravity wave currents found by Chang and Anthes (1978), Shay et al. (1990), and Bender et al. (1993). As discussed by Chang and Anthes (1978) and by Price (1981), the wind stress on the ocean rotates clockwise with time to the right of the storm track and counterclockwise with time to the left of the track, creating near-resonant coupling between this stress and the wind-induced near-inertial rotating current. In Figs. 10 and 11 we compare the distributions of sea surface temperature after 24, 48, 72, and 96 h of integration in the four experiments. In every case, the maximum cooling is located to the right of the track and shifts closer to the track with time as the atmospheric vortex develops. The rear-rightward bias of the SST response to tropical cyclones seen here is a known feature that has been established by observations (Wright 1969; Pudov et al. 1979; Black 1983; Sanford et al. 1987; Shay et al. 1989) and by previous numerical experiments (Chang and Anthes 1978; Price 1981; Khain and Ginis 1991; Bender et al. 1993). This feature is most pronounced in experiment E3, LDN (Fig. 11a). In this case, the maximum decrease in SST reaches 4.1°C by 96 h. It is least pronounced in experiment E4, LUD (Fig. 11b), in which the storm moves into the region of maximum OMLD (some 40 m deeper than at its original location) during the first 48 h, giving sea surface cooling of less than 1°C, and then moves across decreasing OMLD, with the maximum reduction of the SST at the end of 96 h of integration being only 1.8°C. In experiments E1 (CTL) and E2 (LUP), the SST changes are 3.3° and 1.6°C, respectively, by 96 h. It will be noted, too, that the maximum cooling in experiment E2 is confined to the region of shallow mixed layer depth far behind the storm. It is clear from this result that the increasing mixed layer depth in the latter case greatly mitigates the cooling along the path of the TC and that, in general, spatial variations of the OMLD play a significant role in determining the SST response and the resulting feedback on the atmospheric disturbance. It should be noted that the decrease of SST in the wake of a TC and its feedback to the TC are sensitive to both the intensity and the translation speed of the TC. For example, the minimum sea level pressure increases faster in E4 than in E2 in the time window between 75 and 85 h (Fig. 5), because the maximum SST cooling in E4 is more proximate to the TC than that in E2 (Figs. 10b and 11b) as the TCs in both experiments travel into regions of shallower and deeper OMLD, respectively.
In addition to experiment E2, in which the horizontal resolution of the coupled model is set at 40 km, we ran two more experiments by keeping everything else unchanged but varying the horizontal resolution of the coupled model to 30 and 50 km, respectively. Results of both runs and that of E2 (40-km resolution) indicate that a TC simulated in the coupled model with the finer resolution tends to develop to a greater intensity than the ones with a relatively coarse resolution. Sensitivity of the TC tracks to model horizontal resolution suggests that the TC with greater intensity tends to move less northward, whereas the TC with less intensity tends to deflect farther northward (Fig. 12). This appears in agreement qualitatively with the results by Bender et al. (1993).
It is well known that the oceanic mixed layer acts as a heat reservoir that, when sufficiently warm and deep, provides the sensible and latent heat energies to fuel the growth of tropical cyclones traveling over it. It has also been established both theoretically and observationally that there is a strong relationship between sea surface temperature and cyclone intensity (Emanuel 1988; Evans 1993; DeMaria and Kaplan 1994; Whitney and Hobgood 1997). What we set out to do here was to ascertain the role of spatial variations of the OMLD on this process. By taking the SST initially uniform in all of our numerical experiments, we were able to isolate the effects of such variations.
Our results suggest that spatial variations of the initial oceanic mixed layer depth comparable to those observed in the Atlantic and Pacific Oceans have a significant effect on the rate of intensification of tropical disturbances and on the distribution and intensity of the storm-induced sea surface cooling. When the OMLD increases toward the west with a slope characteristic of the monthly mean slope observed in the major oceans, the well-known cooling effect due to entrainment of cooler water into the mixed layer from below is moderated and the TC develops to a greater intensity and maintains this intensity longer than it would otherwise have if the mixed layer were uniform depth. When the OMLD decreases toward the north by a characteristic amount, which is smaller than the typical westward increase, the effect is in the opposite sense. Moreover, when both variations are prescribed simultaneously and the TC is allowed to move first over an increasingly deep and then over a decreasingly deep mixed layer, it develops to greater intensity than it would have over a mixed layer of constant depth. This is because the development takes place during the first part of the excursion over the deepening mixed layer. The differences of maximum wind speed and minimum surface pressure between experiment E3 (in which the OMLD decreases along the path of the tropical disturbance) and experiment E4 (in which it first increases and then decreases) are −3.1 m s−1 and 4.7 hPa, respectively, and the differences in maximum sea surface cooling are 2.3°C. The path of the atmospheric disturbance, however, appears to be insensitive to these differences. A recent study by Hong et al. (2000) on intensification of Hurricane Opal (1995) over a warm core ring in the Gulf of Mexico lends credence to our conclusions about the importance of variations of the ocean subsurface heat content in the intensification of TCs.
Results of our numerical experiments also show that the intensity and motion of TCs are quantitatively sensitive to the changes of model horizontal resolution. The TC simulated with a finer resolution tends to develop to a greater intensity compared to the one with a coarse resolution. The former also tends to deflect less northward than the latter while traveling in an easterly mean flow of a medium speed. This appears to be in agreement, at least qualitatively, with the result of a previous numerical simulation using a different coupled system (Bender et al. 1993).
Our results should be considered as only a first-order approximation to what takes place during the life cycle of a tropical cyclone. The horizontal resolution of the model was not fine enough to characterize the mesoscale dynamics of TCs. The vertical resolution was also coarse. Model physics was kept simple, especially in the planetary boundary layer. Only bulk formulas were used in the evaluation of momentum, and sensible and latent heat fluxes across the air–ocean interface. The ocean model was highly simplified. Salinity was not taken into account in the density variation of the mixed layer. The structure of the thermocline was represented in a fairly simple manner. External waves at the sea surface were eliminated with the use of the rigid-lid approximation. Despite these limitations and simplicities, the present coupled model was able to reflect the primary features in the TC–ocean system and to provide insight into the role of variations of the mixed layer depth in tropical cyclone intensification.
The authors would like to thank Dr. Sydney Levitus for providing the monthly mean oceanic mixed layer depth data, and Dr. Karl H. Bathen for letting us use his original mixed layer depth plots for analysis. Dr. Lynn Keith Shay’s helpful comments on the early version of this paper are appreciated. Comments from three anonymous reviewers greatly improved the quality of this paper. The first and third authors thank the National Science Foundation (NSF) for its support of this research under NSF Grant ATM-9310119 and The Florida State University for making available the required computing time. The second author is supported by NRL/ONR Basic Research Grant PE601153N. Part of the work was done at the University of Alabama in Huntsville and the Tennessee Valley Authority by the first author supported by Grant NSF/NASA-622-01-13.
The Ocean Model
The momentum equations are as follows:
where f = 2Ω sinϕ is the Coriolis parameter and m(ϕ) = [1 + sin(π/3)][1 + sin(ϕ)]−1 is the map scale factor of the polar stereographic projection true at 60°N. The subscripts 1 and 2 designate variables in the ML and TL, respectively. The terms on the right-hand side of Eqs. (A1) and (A2) represent the pressure gradient force, the eddy turbulent diffusion, the momentum transfer by turbulent entrainment through vertical shear of the horizontal currents, and the frictional force, respectively.
The pressure gradient forces in both the ML and the TL are expressed by
where h1 and h2 are the total layer depths of the ML and TL, respectively, and g′1 and g′2 are the reduced gravities in the ML and TL defined by
For simplicity, ρ2 and ρ3 are assumed to be constant, while ρ1 varies with the ML temperature only. On the right-hand side of Eqs. (A3) and (A4), the first term represents the pressure gradient force due to variations of the SST. The second and third terms stand for the pressure gradient force due to the horizontal variations of the OMLD and the TL depth. The ML density ρ1 is computed from the equation of state
Wind stress (τo), and interfacial shear stresses between the ML and the TL (τI) and between the TL and AL (τB) take the following forms:
Numerical values of the various constants and parameters are listed in Table 1.
The continuity equations are as follows:
where η1 and η2 are the perturbations of the ML and TL depths from their mean depths H1(x, y) and H2(x, y), respectively.
The thermodynamic equation is as follows:
where Qo = QS + QL. The sensible and latent heat fluxes are represented by
The subscripts a and 1 designate atmospheric and mixed layer variables, respectively. The last term on the right-hand side of Eq. (A12) represents the cooling of the mixed layer by turbulent entrainment. Here, ϒ(We) is a nonnegative function of the turbulent entrainment rate given by
where the turbulent entrainment rate
Corresponding author address: Dr. Qi Mao, Tennessee Valley Authority, Environmental Research Center, CEB-2A, P.O. Box 1010, Muscle Shoals, AL 35662-1010.
* Geophysical Fluid Dynamics Institute Contribution Number 405.
+ Current affiliation: Tennessee Valley Authority, Environmental Research Center, Muscle Shoals, Alabama.