Restratification of the Upper Ocean after the Passage of a Tropical Cyclone: A Numerical Study

Wei Mei Department of Earth System Science, University of California, Irvine, Irvine, California

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Claudia Pasquero Department of Earth System Science, University of California, Irvine, Irvine, California, and Dipartimento di Scienze Geologiche e Geotecnologiche, Università di Milano-Bicocca, Milano, Italy

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Abstract

The role of baroclinic instability in the restratification of the upper ocean after the passage of a tropical cyclone (TC) is determined by means of numerical simulations. Using a regional ocean model, the Regional Ocean Modeling System (ROMS), a high-resolution three-dimensional simulation that includes the process of baroclinic instability and is initialized with moderate-amplitude eddy structures reproduces the satellite-observed decay rate of the TC-induced sea surface temperature (SST) anomaly and is also in qualitative agreement with published observations after the passage of Hurricane Fabian in 2003 that showed decaying cold and warm anomalies located in the climatological mixed layer (CML) and upper thermocline, respectively. The model ocean is restratified after approximately one month with a net heat gain in the water column due to anomalous air–sea heat fluxes. The model shows, however, that vertical heat fluxes associated with baroclinic instability dominate over air–sea heat fluxes in restoring the CML heat content during the first month. A comparison with two-dimensional simulations that exclude baroclinic adjustment further highlights the importance of baroclinic instability: it can not only input a considerable amount of heat into the CML, but also establish strong stratification there, inhibiting the downward penetration of heat contributed by diabatic heating at the surface; both effects hasten the recovery of the SST.

Additional experiments were performed to examine the sensitivity of the model results to changes in Newtonian cooling rate, changes in the magnitude of the eddy structures used to initialize the simulation, and changes in poststorm wind strength; the results indicate that, although some of them may have a significant effect on the recovery time of the SST, their influence on the contribution of baroclinic instability to the recovery of the CML heat content is modest. However, the contribution of baroclinic instability exhibits pronounced positive dependence on the depth of the mixing layer relative to the CML depth and the relative size of the area with unperturbed water. Its dependence on the shape of the spatial variation of the mixing depth is relatively weak but in a more complicated manner. These dependencies are consistent with those predicted by a simple front adjustment model, whereas the latter also suggest that the contribution of baroclinic instability is independent of the prestorm stratification below the CML.

Overall, the idealized simulations in this study suggest that, for a typical situation in the real ocean, baroclinic instability can account for approximately 50% of the full recovery of the CML heat content, whereas under specific conditions the contribution can be significantly smaller. Those estimates provide a limit to the maximum net warming of the water column after the initial mixing event and thus have important implications regarding estimating the long-term effect of TCs on the upper-ocean heat budget.

Corresponding author address: Wei Mei, Department of Earth System Science, University of California, Irvine, Irvine, CA 92697. E-mail: meiw@uci.edu

Abstract

The role of baroclinic instability in the restratification of the upper ocean after the passage of a tropical cyclone (TC) is determined by means of numerical simulations. Using a regional ocean model, the Regional Ocean Modeling System (ROMS), a high-resolution three-dimensional simulation that includes the process of baroclinic instability and is initialized with moderate-amplitude eddy structures reproduces the satellite-observed decay rate of the TC-induced sea surface temperature (SST) anomaly and is also in qualitative agreement with published observations after the passage of Hurricane Fabian in 2003 that showed decaying cold and warm anomalies located in the climatological mixed layer (CML) and upper thermocline, respectively. The model ocean is restratified after approximately one month with a net heat gain in the water column due to anomalous air–sea heat fluxes. The model shows, however, that vertical heat fluxes associated with baroclinic instability dominate over air–sea heat fluxes in restoring the CML heat content during the first month. A comparison with two-dimensional simulations that exclude baroclinic adjustment further highlights the importance of baroclinic instability: it can not only input a considerable amount of heat into the CML, but also establish strong stratification there, inhibiting the downward penetration of heat contributed by diabatic heating at the surface; both effects hasten the recovery of the SST.

Additional experiments were performed to examine the sensitivity of the model results to changes in Newtonian cooling rate, changes in the magnitude of the eddy structures used to initialize the simulation, and changes in poststorm wind strength; the results indicate that, although some of them may have a significant effect on the recovery time of the SST, their influence on the contribution of baroclinic instability to the recovery of the CML heat content is modest. However, the contribution of baroclinic instability exhibits pronounced positive dependence on the depth of the mixing layer relative to the CML depth and the relative size of the area with unperturbed water. Its dependence on the shape of the spatial variation of the mixing depth is relatively weak but in a more complicated manner. These dependencies are consistent with those predicted by a simple front adjustment model, whereas the latter also suggest that the contribution of baroclinic instability is independent of the prestorm stratification below the CML.

Overall, the idealized simulations in this study suggest that, for a typical situation in the real ocean, baroclinic instability can account for approximately 50% of the full recovery of the CML heat content, whereas under specific conditions the contribution can be significantly smaller. Those estimates provide a limit to the maximum net warming of the water column after the initial mixing event and thus have important implications regarding estimating the long-term effect of TCs on the upper-ocean heat budget.

Corresponding author address: Wei Mei, Department of Earth System Science, University of California, Irvine, Irvine, CA 92697. E-mail: meiw@uci.edu

1. Introduction

It has been well documented that the passage of a tropical cyclone (TC) often generates a cold anomaly in the prestorm mixed layer, which for simplicity we refer to as the climatological mixed layer (CML), and a warm anomaly in the upper climatological thermocline (e.g., Price 1981; Brooks 1983; Pudov 1993; Zedler et al. 2002). Meanwhile, the heat content of the water column is approximately conserved as the observed thermal anomalies are mainly a result of the heat redistribution in the vertical by intense shear-induced mixing (e.g., Price et al. 1994; Jacob et al. 2000; Black et al. 2007; Sanford et al. 2007; Huang et al. 2009).

Usually, the surface cold anomaly is observed to disappear within several weeks after the TC passage (Hazelworth 1968; Nelson 1996; Hart et al. 2007; Price et al. 2008, hereafter PMN08; Lévy et al. 2012). The warming rate can differ from case to case because of the differences in environmental factors such as wind speed and cloud cover, but the sea surface temperature (SST) anomaly (SSTA) usually disappears with an e-folding time of 5–20 days, as has been determined from satellite and drifter data by PMN08. However, unlike the surface counterpart, the positive temperature anomaly in the subsurface layer has been postulated to persist much longer (Pasquero and Emanuel 2008). As a result, the upper ocean may experience a net heat gain after the TC passage, when anomalous air–sea heat fluxes contribute to the removal of the surface cold anomaly.

The potentially increased heat content in the upper ocean has significant effects both locally and globally. First, the persistent warm subsurface anomaly may remain intact and persist in the seasonal thermocline and thereby provide more energy for the development and strengthening of subsequent TCs (Shay et al. 2000) and may eventually be erased during the coming winter season because of the deepened mixed layer (Namias and Born 1970, 1974; Pudov 1993; Alexander and Deser 1995; Alexander and Penland 1996; Jansen et al. 2010). Second, it has been suggested that the net warming of the upper ocean is balanced by meridional heat transport out of the subtropical regions and given up to the atmosphere at different latitudes (e.g., Emanuel 2001; Jayne and Marotzke 2001). For example, Emanuel (2001) and Sriver and Huber (2007) suggest that the net ocean heating associated with TC activity could explain an appreciable part of the observed poleward ocean heat transport, whereas more recent numerical work by Pasquero and Emanuel (2008), Jansen and Ferrari (2009), and Fedorov et al. (2010) show that the TC-induced mixing and upper-ocean warming lead to an increased equatorward heat transport and thus shape tropical climate through the shallow subtropical cell. Whether TCs eventually affect the poleward or equatorward ocean heat transport is still under debate (Emanuel 2002; Korty et al. 2008; Jansen and Ferrari 2009), but what is clear is that a precise estimate of changes in upper-ocean heat content induced by TCs is beneficial and even crucial to hurricane forecasting as well as climate predictions over a variety of time scales (e.g., Smith et al. 2007; Dunstone and Smith 2010).

The first step for such an accurate assessment of changes in the upper-ocean heat content is a detailed study of the restratification process in the upper ocean after the TC passage. However, previous work addressing the decay of the cold wake and the impact of TCs on meridional heat transport is based on an idealized and simplified assumption that the cold anomaly in the CML is completely removed by anomalous air–sea heat fluxes (e.g., Emanuel 2001; Sriver and Huber 2007). This is quite different from the real situation because adiabatic processes within the upper ocean also contribute to the restoration of normal conditions. These unconsidered processes, at least, include vertical heat rearrangement associated with baroclinic instability and large-scale horizontal heat advection. In this paper, we focus on the former, recognizing that the latter may also contribute significantly.

Because of the large SST reduction within the cold wake, strong temperature gradients form between the center of the wake and the surrounding unperturbed seawater. For instance, Black et al. (2007) report that SST differences up to 2°C exist over a 50-km distance at the leading edge of the cold wake left behind Hurricane Frances (2003). Such thermal fronts are often unstable for different reasons, including baroclinic instability. Baroclinic instability induces cross-front ageostrophic circulation cells with upwelling in the lighter-water side and downwelling in the denser side, resulting in an upward buoyancy flux and thereby restratifying the upper ocean. This process can account for a large part of the heat budget in the surface mixed layer (Rudnick 1996; Boccaletti et al. 2007; Capet et al. 2008a,b; Thomas et al. 2008). It is worth noting that, in the last few years, considerable attention has been devoted to the submesoscale baroclinic instability (or frontal instability) instead of the conventional mesoscale baroclinic instability. The submesoscale instability, which is observed to occur along the near-surface density fronts (e.g., Garvine et al. 1988), is surface trapped and has a spatial scale of several kilometers and a growth time scale of a couple of days. Because of the fast growth rate, the restratification associated with the submesoscale instability is rapid enough to compete with processes such as wind-induced mixing that destroy the stratification. Moreover, numerous numerical studies have shown that submesoscale instability is considerably stronger than mesoscale baroclinic instability in terms of its role in the restratification in the upper ocean by exchanging heat and salinity in the vertical (e.g., Boccaletti et al. 2007).

Dynamical processes, such as submesoscale instability, and thermodynamical processes, such as air–sea heat fluxes, work together during the upper-ocean restratification, with their relative importance being time dependent. Taking these processes into account can provide a more accurate and realistic estimate of the heat gain by the upper ocean due to the passage of TCs. The goal of this study is to investigate the restratification of the upper ocean after the passage of a TC, with particular emphasis on the role of baroclinic instability, by means of numerical simulations. Specifically, we first obtain a general idea of the maximum possible contribution of baroclinic instability to the restoration of the CML heat content using a simple front adjustment model. Then, a primitive equation ocean model including diabatic and diffusive processes is employed to explore the restratification process in detail and to quantify the relative importance and contribution of those above-mentioned competing processes in setting a new equilibrium state. Furthermore, additional experiments are performed to test the sensitivity of the numerical results to magnitude and spatial structure of the initial temperature anomaly, external forcing strength, relative depth of the mixing layer, and extension of model domain.

2. A simple front adjustment model

We first estimate the maximum possible contribution of baroclinic instability to the recovery of the CML heat content using a simple front adjustment model. The dashed curve in Fig. 1 shows the vertical climatological temperature profile with a mixed layer depth of hcml and a stratification below measured as the vertical temperature gradient with a value of Γ. This climatological profile is then subject to both diabatic cooling at the air–sea interface and adiabatic mixing within the water column associated with the TC passage, resulting in a new profile with a mixing layer depth of h and a reduced SST (solid curve in Fig. 1). (We will refer to this new profile as the initial state of the restratification process.) If we assume there is no stratification in both the CML and the new mixing layer, then the deviation of the SST (ΔSST) from its climatological value T0 and the new mixing layer depth h can be related using a simplified version of upper-ocean heat balance,
e1
where t is time, Q is the net air–sea heat flux, cp is the specific heat of seawater, and ρ0 is the seawater density. Because the contribution of net air–sea heat fluxes to the SST or surface layer cooling is rather small compared to the contribution of the vertical mixing and entrainment within the upper ocean produced by the wind- and wave-generated and shear-induced turbulence (e.g., Price 1981; D’Asaro et al. 2007), we assume Q = 0 in Eq. (1). Then we have
e2
Fig. 1.
Fig. 1.

Schematic vertical temperature profiles associated with the passage of a TC. The dashed line shows the temperature profile of a climatological state with a mixed layer depth of hcml, and the solid line is the temperature profile with a mixing layer depth of h after the effect of wind stress. The reduction in the SST corresponding to such a deepening is .

Citation: Journal of Physical Oceanography 42, 9; 10.1175/JPO-D-11-0209.1

In reality, the wind-induced mixing depth h is spatially nonuniform. In this study, its spatial variation is approximated using the following function (h is assumed to be constant along the cold wake, which is defined as in the zonal direction, i.e., the x direction):
e3
where y is the meridional coordinate; hmax, representing the induced maximum mixing depth associated with the TC passage, is the mixing depth at y = Ly; k is a shape parameter whose magnitude sets the width and thus the strength of the temperature front between y = 0 and y = Ly; and n is a parameter that determines the relative extension of the unperturbed water (i.e., water characterized with climatological temperatures). Figure 2 displays three cases we shall consider as initial states for the restratification process, with Figs. 2a–c showing kLy→+∞ m, kLy = 10 m, and kLy→0 m, respectively.
Fig. 2.
Fig. 2.

Cross section of the temperature profiles used in the simple front adjustment model with (a) kLy→+∞ m, (b) kLy = 10 m, and (c) kLy→0 m in Eq. (3). The gray scale on the bottom applies to all subplots. Contours are at 22°, 24°, 26°, 27°, and 28°C.

Citation: Journal of Physical Oceanography 42, 9; 10.1175/JPO-D-11-0209.1

Because the net effect of baroclinic instability working on a front is to slump the isopycnals from the vertical to the horizontal, the final state should be vertically stably stratified. Accordingly, the basic idea of the simple front adjustment model presented here is to rearrange the water masses according to their temperatures until a barotropic stratification is obtained. Processes including thermal diffusion and air–sea heat fluxes are excluded. Typically, after this adjustment the CML is on average colder than before the formation of the front. The remaining warming of the CML to reach climatological conditions (not attained in this simple model) must be associated with diabatic processes and/or large-scale heat advection. Using temperatures of initial perturbed state and of the final adjusted barotropic state in the CML together with their climatological values, we can calculate the contribution of baroclinic instability in recovering the CML heat content. We can get analytical solutions to special cases like those shown in Figs. 2a,c, whereas for a general case with any value of kLy it is easy to solve numerically. For briefness, here we only present the main conclusions from this simple model with the detailed analytical solution to the case in Fig. 2a given in appendix A.

The basic features in the three cases are similar (Fig. 3): the heat content in the CML at the end of the restratification process is more similar to the climatological heat content if the relative area of unperturbed water n (see appendix A) is larger and/or the normalized mixing depth hmax/hcml is larger. In those situations, baroclinic instability contributes to most or all the CML warming necessary to reach climatological conditions. Note that the contribution is independent of the specific depth of the CML and of the mixing depth: it only depends on their ratio.

Fig. 3.
Fig. 3.

Dependence of the contribution of baroclinic instability to the recovery of the CML heat content r on the wind-induced deepening normalized by hcml and on the extension of the unperturbed water normalized by Ly for the three situations, (a) kLy→+∞ m, (b) kLy = 10 m, and (c) kLy→0 m, shown in Fig. 2.

Citation: Journal of Physical Oceanography 42, 9; 10.1175/JPO-D-11-0209.1

The dependence of results on the value of kLy is more complicated (Figs. 4a,b): when n or hmax/hcml is small, baroclinic instability contributes more for smaller kLy, but it makes less contribution for smaller kLy when n or hmax/hcml is large. One interesting point from this simple model is that the contribution of baroclinic instability is independent of the stratification below the CML [i.e., Γ; Fig. 4c and Eq. (A10)].

Fig. 4.
Fig. 4.

Dependence of the contribution of baroclinic instability to the recovery of the CML heat content r (a) on k and the wind-induced deepening normalized by hcml with no extension of unperturbed water and the thermocline stratification Γ = 0.04°C m−1, (b) on k and the extension of the unperturbed water normalized by Ly with hmax = 2.62hcml and Γ = 0.04°C m−1, and (c) on k and Γ with hmax = 2.62hcml and no extension of unperturbed water.

Citation: Journal of Physical Oceanography 42, 9; 10.1175/JPO-D-11-0209.1

Although analysis of this simple model provides us with a general idea on the magnitude of the contribution of baroclinic instability and its dependence on other parameters, it cannot serve for a process study. Thus, we will employ a primitive equation ocean model including diabatic and diffusive processes to investigate the restratification process of the upper ocean in detail. The results from the primitive equation model will be quantitatively compared with those obtained in this part.

3. Primitive equation ocean model and experimental designs

We solve the primitive equations in an idealized channel configuration on an f plane representing an open ocean area away from continents, subject to a (spatially uniform and temporally variable) wind stress forcing and a heat flux forcing that resolves the diurnal cycle. The domain of size Lx × Ly × H has periodic east–west lateral boundary conditions and wall boundaries at the northern and southern edges. Side and bottom boundary conditions are free slip. Salinity is kept uniform in the domain and does not vary with time, so that density depends on temperature only, through a linear equation of state. Because the focus in this study is not on the mixing induced by the passage of the TC but rather on the restratification process that takes place in the following days, in this domain we initialize the model with an idealized front that mimics the edge of the cold wake left by the TC passage as those shown in Fig. 2 and study its simulated evolution.1 Consistently, the wind stress imposed at the surface does not refer to TC wind intensities but rather to the typical wind conditions in subtropical regions away from strong disturbances.

The wind stress is stochastic in time and homogeneous in space over the whole domain. Its temporal evolution is defined by a Langevin equation (see Rodean 1996),
e4
where τ refers to either zonal or meridional wind stress, στ is its standard deviation, t is time, tL is the autocorrelation time scale, and ξ(t) is a Gaussian random series with zero mean and variance 1. This equation has been widely used to generate stochastic atmospheric forcing in modeling studies of the upper ocean (e.g., Alexander and Penland 1996; McWilliams et al. 2009). The values of tL and στ are chosen to obtain a wind stress with characteristics similar to the climatologies (see Table 1).
Table 1.

Parameter values for the presimulation and the reference run of the restratification simulation.

Table 1.
The solar radiative flux Qsw in our simulations is expressed as the following function:2
e5
where Qd is the daytime heating parameter and th is the length of daytime heating. The water type, which determines the penetrating depth of shortwave radiation, is chosen as IA (Jerlov 1976). The heat loss from the ocean surface Qloss (i.e., the sum of longwave radiation and latent and sensible heat flux) is combined in a temperature restoring term,3
e6
where λ is the Newtonian cooling rate, T0 is a reference temperature, and Ts is the surface temperature. When the ocean temperature is in equilibrium, (an overbar denotes temporal average over a day), no net heat flux enters the air–sea interface. Then, we have and thus .

The model is run for a presimulation, starting from a resting linearly stratified ocean with T(z) = Teq + Γz, to obtain a quasi-steady state, characterized by a well-developed mixed layer of depth hcml (the simulated CML).4 Then, using the obtained steady-state temperature profile, an artificial front is created similar to those shown in Fig. 2: the TC-induced mixing layer depth varies with y (i.e., the meridional direction) and is modeled using Eq. (3) with n = 0; the values of k, Ly, and hmax are given in Table 1. The temperature of the mixing layer at any latitude is obtained by imposing conservation of heat content over the water column down to the mixing depth, similar to that in the simple model. The resulting mixing layer temperature is a monotonic function of latitude with the lowest temperature found in correspondence of the deepest mixing layer on the northernmost side of the domain (see Fig. 5). The same temperature distribution as a function of latitude and depth applies to all longitudes and thus is uniform in the zonal direction. This resting zonally uniform front develops into a jet according to the thermal wind relation. The jet is baroclinically unstable and we need to seed the initial field with some perturbation to let the instabilities grow. We thus superpose to the thermal front a temperature field that mimics the turbulent eddy structure present in the real ocean. Details for generating the initial disturbance are given in appendix B. The resulting field is then used as initial condition for the restratification simulations.

Fig. 5.
Fig. 5.

Meridional section of the initial temperature front for the restratification simulation (°C; contours are at 15°, 20°, 25°, 26°, 27° and 28°C). On this temperature front, eddy-like perturbations are added to generate the initial conditions for the restratification simulation (see text for details). Mixed layer depth varies between hcml = 76.26 m, which defines the simulated CML depth, and hmax = 200 m, which is a reasonable mixing depth induced by slow-moving TCs [see Brooks (1983) for an observational example and Sriver (2010) for a modeling example]. The simulated climatological thermocline extends from the base of the CML to the model bottom (i.e., between 76.26 and 500 m). The surface temperatures at the southernmost and the northernmost points are 28.67° and 25.89°C, respectively, resulting in a maximum SSTA of −2.78°C. Used parameter values are listed in Table 1.

Citation: Journal of Physical Oceanography 42, 9; 10.1175/JPO-D-11-0209.1

The model used is the Regional Ocean Modeling System (ROMS; Shchepetkin and McWilliams 2005). We use the hydrostatic version of the code because the average vertical fluxes are not significantly different between hydrostatic and nonhydrostatic cases (Mahadevan 2006). In all experiments, third-order upstream bias horizontal advection is used for both temperature and momentum, and fourth-order centered vertical advection is used for temperature. Harmonic horizontal mixing is used for both temperature and momentum. The nonlocal K-profile parameterization (KPP) scheme (Large et al. 1994) is chosen to parameterize the vertical turbulent mixing. We tested the sensitivity of our model results to the choice of the parameterization schemes of the vertical mixing by repeating the reference restratification simulation in section 4 utilizing the Mellor–Yamada-2.5 scheme and the generic length scale methods individually. We found no significant differences.

Because the submesoscale processes are important candidates for the upper-ocean restratification and exhibit spatial scale of a few kilometers, the experiments are performed with a horizontal resolution of 1 km × 1 km. The model has 200 unequally spaced σ layers in the vertical with 115 layers above 100 m. All simulations are 150 days long, using a time step of 30 s. The model output is sampled either twice per day or eight times per day to resolve the diurnal cycle.

4. Reference experiment

The temporal evolution of the initial front (Fig. 5) described in the previous section is presented here. We refer to the reference simulation for the parameter values listed in Table 1, where the model is subject to the same wind and heat forcing used for the presimulation and representing normal weather conditions after the TC passage. We examine how SST, potential temperature at depths, upper-ocean stratification, and CML heat content evolve, with the relative importance of anomalous air–sea heat fluxes and adiabatic vertical heat fluxes associated with baroclinic instability being emphasized. Note that both the prestorm mixed layer depth hcml and the mixing depth hmax are relatively large in this simulation, considering that similar values are observed only in extreme situations. However, as will be shown in section 5c, the results are quite insensitive to the precise values of those parameters and only depend on their ratio, which here is hmax/hcml = 2.62 and thus corresponds to a common situation.

a. Evolution of the SST

The temporal evolution of the domain-averaged SSTA with respect to the climatological condition is shown in Fig. 6. Also shown is the temporal evolution of the area-mean composite SSTA associated with the passage of TCs of hurricane intensity obtained from satellite data [thick dashed curve in Fig. 6; data and methods for producing this curve are summarized in appendix C with more details given in Mei and Pasquero (2012, manuscript submitted to J. Climate)]. All the SSTAs are normalized by their respective maximum cooling: 1.27°C for the model results and 1.22°C for the observations. Similar to that in the observations, the adjustment of the SSTA in the model takes the form of a roughly exponential relaxation: it decays rapidly during the first week and then diminishes slowly. Besides the apparent positive trend, the model SSTA also shows strong diurnal variations, which can exceed 0.5°C. This is consistent with the results revealed by the drifter data in PMN08 (see their Fig. 4b). The general warming rate deduced from the model SSTA evolution matches reasonably well with the composite satellite observations. The e-folding time obtained using an exponential fit is about 9 days. The relatively larger normalized SSTA after day 25 in our model compared with the normalized SSTA in the observations may be due to the absence of large-scale horizontal advection or the limited area of water characterized with climatological temperatures in our simulation.

Fig. 6.
Fig. 6.

Temporal evolution of scaled SSTA in the reference run of the restratification simulation (thin solid curve) and of composite SSTA associated with the passage of TCs of hurricane intensity based on TMI observations (thick dashed curve; data and methods for producing this curve are given in appendix C). They are scaled by their respective maximum cooling: 1.27°C for the numerical simulation and 1.22°C for the observations. Wiggles in the model run are the diurnal cycle. Observational sampling does not resolve the diurnal cycle.

Citation: Journal of Physical Oceanography 42, 9; 10.1175/JPO-D-11-0209.1

b. Evolution of temperatures at depth

Because the temperature at depths cannot be directly observed by satellites, here we use measurements from the Bermuda Testbed Mooring (BTM) taken in 2003 near the center of the cold wake of Hurricane Fabian to validate the model. The weather was reported to be fair for the first weak after the hurricane passage (Black and Dickey 2008; PMN08). We shall compare the temperature averaged over the northern 2/5 (i.e., y ≥ 60 km in Fig. 5) of the model domain with the BTM measurements. This comparison can only be qualitative because of the idealized nature of our simulation, which is meant to represent a general situation and not a specific case. Note that we are only interested in comparing the temporal and spatial structure of the anomalies and not the magnitude of the anomalies, which is known to vary significantly based on TC intensity, translation speed, and upper-ocean conditions (e.g., Price 1981; Sakaida et al. 1998; Mei et al. 2012). For reference, we first show the changes in the vertical temperature profile caused by the passage of the hurricane (Fig. 7). The CML in the model is about 4 times as deep as that before the passage of Hurricane Fabian. The model water in the CML cools by about 2.5°C, whereas the observed temperature drops by a larger value of 3.2°C. Concurrent warming occurs between 40 and 100 m in the Fabian case and is located between 110 and 200 m in the model. Because of those differences in the setup (i.e., the initial conditions before the restratification process starts, which in the numerical simulation were not chosen to reproduce the observations, as we stated above), we shall compare the qualitative evolution of the anomalies.

Fig. 7.
Fig. 7.

(a) Climatological (dashed curve) and poststorm (solid curve) temperature profiles averaged over the northern 2/5 (i.e., y ≥ 60 km in Fig. 5) of the model domain in the restratification simulation. (b) BTM measurements of prestorm (dashed curve) and poststorm (solid curve) temperature profiles associated with Hurricane Fabian in 2003 (from Black and Dickey 2008).

Citation: Journal of Physical Oceanography 42, 9; 10.1175/JPO-D-11-0209.1

The modeled evolution of temperature in the CML after the TC passage is similar to that of the SST, except that the warming rate decreases with depth (Figs. 8a,c). At the same time, the warm anomaly in the upper climatological thermocline tends to weaken gradually, and its cooling rate is generally proportional to its magnitude. Such warming and cooling tendencies are in good agreement with the observations (Figs. 8b,d), suggesting that our model includes the processes principally responsible for these features. To address the possibility that large-scale horizontal advection might have led to the same results, we note that the study area we have chosen is not located in the region with strong horizontal currents and that the tracks of the surface drifters shown in Fig. 4a of PMN08 are along the cold wake of Hurricane Fabian instead of across it. Advection along the cold wake cannot efficiently remove the thermal anomalies because of the weak temperature gradients in that direction. We also note that, if the horizontal advection were to dominate the removal of the thermal anomaly in the upper ocean, then we should observe the decay of temperature anomaly at all depths. We find, however, that the cold anomaly right at the base of the CML is sustained for as long as a week before warming up (see the evolution of the temperature at 35-m depth in Fig. 8d), whereas the temperature near the surface starts increasing within the first 2 days after the passage of the TC. The same phenomenon is also produced in our simulation (see the evolution of temperatures at depths of 50 and 100 m in Fig. 8c). At least for the case of Hurricane Fabian, we may conclude that the large-scale horizontal advection is negligible for the removal of the thermal anomalies in the upper ocean. In other situations, horizontal advection could play a larger role.

Fig. 8.
Fig. 8.

(a) Evolution of vertical temperature profile averaged over the northern 2/5 (i.e., y ≥ 60 km in Fig. 5) of the model domain in the reference run of the restratification simulation (units: °C). (b) As in (a), but for the BTM measurements after the passage of Hurricane Fabian in 2003 (from Black and Dickey 2008). (c) Time series of temperature at different depths averaged over the northern 2/5 (i.e., y ≥ 60 km in Fig. 5) of the model domain in the reference run of the restratification simulation. (d) As in (c), but for the BTM measurements during and after the passage of Hurricane Fabian in 2003 (from Black and Dickey 2008). Dates in (b) and (d) are relative to 5 Sep 2003. Dashed lines in (c) represent temperatures in steady state for day −1 and changes in temperature due to the passage of a TC that are assumed to be linear during day 0. Note the different depth range in the ordinate axis in (a) and (b).

Citation: Journal of Physical Oceanography 42, 9; 10.1175/JPO-D-11-0209.1

After one month, the initial strong temperature front in the model has disappeared, the isopycnals are flat, and strong stratification has built up again in the upper climatological thermocline (Fig. 9). However, a vertical dipole of temperature anomaly remains in the upper ocean. Vertically, however, both the domain-averaged warm and cold anomalies are more concentrated than the initial state because of their horizontal spreading, and both their centers are getting closer to the base of the CML.

Fig. 9.
Fig. 9.

Vertical section of (a) daily averaged zonal-mean temperature on day 30 and (b) the corresponding anomaly with respect to the climatological state in the reference run of the restratification simulation (units: °C).

Citation: Journal of Physical Oceanography 42, 9; 10.1175/JPO-D-11-0209.1

c. Evolution of the heat content

The above analysis shows that both the cold and warm anomalies located respectively in the CML and the upper thermocline decay with time. Here, we give a quantitative description of the temporal change in the heat content of these two layers (Fig. 10a). The CML heat content exhibits a rapid increase during the first 20 days, followed by a slower warming, whereas the climatological thermocline undergoes a quick cooling initially and remains nearly constant afterward. Overall, the total water column has experienced a net warming of 1.63 × 108 J m−2 within the first 3 months, out of a maximum possible of 3.96 × 108 J m−2 that would have occurred if the conditions of the CML went back to the climatological state and there were no heat fluxes at the base of the CML.

Fig. 10.
Fig. 10.

(a) Changes in heat content of the CML (solid black curve), of the upper climatological thermocline (dashed black curve), and of the total water column (solid gray curve). (b) Cumulative vertical heat flux into the CML at the surface (solid curve) and at the base of the CML (dashed curve) scaled by the total amount of heat that is needed to completely restore the CML heat content in the reference run.

Citation: Journal of Physical Oceanography 42, 9; 10.1175/JPO-D-11-0209.1

Now we consider the relative importance of different processes that contribute to the restratification using the CML heat balance averaged over the full horizontal extent of the domain,
e7
where w is the vertical velocity and the angular bracket denotes an average over the x and y plane and a tilde is a departure from that average. The term on the left-hand side is the rate of change of the CML heat content. The first term on the right-hand side is the vertical heat flux at the base of the CML. We refer to this term as the contribution of baroclinic instability as its induced heat flux is one order larger than that associated with geostrophic adjustment and wind-induced upwelling and downwelling. The remaining two terms are the net heat flux at the surface minus the portion of solar irradiance penetrating underneath the CML and the turbulent mixing term at the base of the CML, respectively.

In the model, two processes are principally responsible for the removal of the thermal anomalies: anomalous air–sea heat fluxes and vertical heat fluxes associated with baroclinic instability. Their respective cumulative contributions to the recovery of the CML heat content as a function of time are shown in Fig. 10b. During the first 5 days, they are of nearly equal importance. Beyond 5 days, however, baroclinic instability makes a greater contribution, and during the first 20 days it is responsible for approximately 2/3 of the warming (28% for baroclinic instability versus 15% for anomalous air–sea heat fluxes).

To illustrate the effect of baroclinic instability, snapshots of temperature and vertical velocity near the surface and at the base of the CML around the end of day 5 are shown in Fig. 11. In the temperature field, large meanders with a wavelength around 50 km have already developed along the initially prescribed front. However, the most striking feature is the narrow cold filaments with a spatial scale of several kilometers in between the relatively larger-scale waves. These submesoscale thermal structures are often results of large-scale deformation fields. Also, a cyclonic cold-core ring with a diameter of 10 km has formed and pinched off, as a result of the backward wave breaking when the plumes of warmer water protrude northward and curl backward opposite the direction of the mean flow. These features are similar to those captured by satellite Advanced Very High Resolution Radiometer (AVHRR) infrared images (e.g., Fig. 3 of Kosro et al. 1991). To offset the squeezed thermal structures, ageostrophic circulations are generated, according to the classic frontogenesis theory. The resulting ageostrophic flow produces intense submesoscale vertical velocities in the process of restoring the geostrophic balance (Figs. 11c,d). The amplitudes of the vertical velocities exhibit strong asymmetry with strong downwelling motions and relatively weaker upwelling, which is compensated by the fact that the upwelling covers larger area than downwelling. Another prominent feature is that the vertical motion is stronger near the base of the CML with a maximum instantaneous speed of ~0.35 cm s−1 (~300 m day−1), whereas near the surface it is hard for the vertical velocity to exceed 0.1 cm s−1. Such large-amplitude vertical velocities generally agree with those reported in previous observational and numerical studies (e.g., Allen and Smeed 1996; Mahadevan and Tandon 2006).

Fig. 11.
Fig. 11.

Snapshots of (a) SST, (b) temperature at the base of the CML, (c) vertical velocity at 5-m depth, and (d) vertical velocity at the base of the CML on day 4.75 in the reference run. The units in (a) and (b) are °C and in (c) and (d) are m day−1. Note the different grayscale in the different panels.

Citation: Journal of Physical Oceanography 42, 9; 10.1175/JPO-D-11-0209.1

As expected, the vertical velocities are in phase with the temperature field: warmer water tends to rise, whereas cooler water tends to sink. The covariance of temperature anomaly and vertical velocity results in a net upward heat transfer, which often exceeds 200 W m−2, comparable with the magnitude of anomalous air–sea heat fluxes. Furthermore, these dynamical heat fluxes warm the surface layer at the expense of cooling the layer below, resulting in an (on average) upgradient heat flux and a double increase in vertical temperature gradient. Accordingly, such a dynamical process is more efficient than diabatic heating from above in terms of building up stratification.

5. Sensitivity experiments

a. Sensitivity to the amplitude of initial temperature disturbance and of the Newtonian cooling rate λ

Either initial temperature fields with different eddy amplitudes or different magnitudes of the Newtonian cooling rate λ may result in different conclusions about the temperature evolution and heat content recovery. Thus, six sets of sensitivity experiments initialized with eddy temperature structures of different amplitudes were carried out. Their initial eddy fields were taken from days 2, 6, 10, 13, 15, and 16 of an unforced experiment (see appendix B), and these six sets of experiments are referred to as ExpSets 1–6, respectively. In addition, a set of experiments excluding baroclinic instability by using a two-dimensional (2D) ROMS (denoted as ExpSet 0)5 were also run to isolate the effect of anomalous air–sea heat fluxes from that of baroclinic processes (keeping in mind that the total heat flux is not the linear superposition of the two processes, because of the nonlinearities and interactions between them). Each set of these experiments (ExpSets 0–6) was run with varying the amplitude of λ over the range 15–150 W m−2 K−1 in increments of 15 W m−2 K−1. The reference run corresponds to ExpSet 5 with λ = 60 W m−2 K−1.

Figures 12 and 13 display the e-folding time of the SSTA decay, recovery of the CML heat content, and contributions of baroclinic instability and anomalous air–sea heat fluxes as a function of the magnitude of the initial eddy structures and λ. Generally, the larger the initial eddies, the less time it takes for the SST to recover. This is because the eddy structure with small amplitude needs more time to develop (the initial linear growth phase of the anomalies lasts longer) and thus slows down the SST restoration, giving support to our claim of the importance of baroclinic instability (note that, in the observations, the SST recovery time is on the order of days). A comparison between the 2D and 3D simulations further demonstrates this implication, especially when λ is relatively small. The e-folding time varies between 150 and 8 days over the entire range of λ displayed in the 2D simulations, whereas this range reduces to be between 12 and 4 days for ExpSet 6 runs.

Fig. 12.
Fig. 12.

(a) The e-folding time (days) obtained using the exponential fit from the sensitivity experiments with different Newtonian cooling rates λ and with different magnitudes of initial eddy structures. Listed numbers are results for ExpSet 0 runs, which are 2D simulations that exclude baroclinic instability. Gray shading and contours are for ExpSet 1–6 runs, which are 3D full simulations. Contours are at 5, 6, 7, 8, 9, 10, 12, 15, and 20 days. The magnitudes of the initial eddy structures in ExpSet 1–6 runs are 10−4°, 10−3°, 10−2°, 0.04°, 0.14°, and 0.22°C, respectively (see appendix B for details). (b) The e-folding time as a function of the Newtonian cooling rate λ for ExpSet 0 run (circles) and for ExpSet 4 run (asterisks). The solid and dashed lines are their respective function fits.

Citation: Journal of Physical Oceanography 42, 9; 10.1175/JPO-D-11-0209.1

Fig. 13.
Fig. 13.

(a) As in Fig. 12a, but for the percent of heat content that has recovered in the CML after 45 days. (b) As in (a), but for the contribution of the vertical dynamical heat flux at the base of the CML. (c) As in (a), but for the contribution of the diabatic heat flux at the air–sea interface. The contour interval is 4%. The grayscale applies to all panels.

Citation: Journal of Physical Oceanography 42, 9; 10.1175/JPO-D-11-0209.1

While baroclinic instability greatly reduces the SST recovery time, it also adds a large amount of heat into the CML. In the 3D simulations, baroclinic instability contributes 20%–30% of the heat content that is needed to fully restore the CML, depending on the initial amplitude of the eddy field. Such a dependence is due to the competition between baroclinic processes and anomalous air–sea heat fluxes. In experiments with smaller eddies, during their development, air–sea heat flux anomaly reduces the front strength and decreases the available potential energy because of the anomalous differential diabatic heating across the front, and thus the contribution of baroclinic instability is smaller. In contrast, in experiments with larger eddies, baroclinic adjustment can rapidly remove the SSTA, contributing more while greatly reducing the contribution of air–sea heat fluxes. This is more prominent when ExpSets 6 and 0 are compared: the heat attributable to anomalous air–sea heat fluxes in ExpSet 6 is 8%–16% less than that in ExpSet 0 after 45 days. Despite this, the overall effect of including baroclinic instability is to promote the recovery of the CML heat content. After 45 days, the CML heat content in the 3D simulations recovers 10%–20% more than in the 2D cases (Fig. 13a).

Based on the above results, it is clear that, through tilting the isopycnals from the vertical to the horizontal, baroclinic instability not only directly adds heat to the CML and thus warms the surface water but increases vertical stratification and thereby establishes a barrier in the CML. This barrier further helps to inhibit the downward heat penetration and limit the vertical extent of heat contributed by anomalous air–sea heat fluxes, resulting in a much faster warming of the surface water and SST, which in turn reduces the contribution of anomalous air–sea heat fluxes.

In both 2D and 3D experiments, increasing λ inputs heat more quickly into the surface layer and thereby speeds up the SST recovery (Fig. 12). For 2D simulations, such a dependence can be described accurately using a power law with a scaling exponent of −1.168, in good accord with PMN08, who find that the e-folding time is inversely proportional to the value of λ. This is to be expected and can be attributed to the assumption that the anomalous air–sea heat flux is the product of λ and SSTA. The small discrepancy between our result and that of PMN08 may be due to the fact that the trapping depth depends on λ in our model. In contrast, the relation between the SST recovery time and λ is more or less linear for 3D experiments, and it is much weaker than the 2D cases. This is not surprising because a large part of the SSTA decay is attributable to baroclinic instability, which fluxes heat up, hence reducing the role of anomalous air–sea heat fluxes. Similar to the e-folding time of the SST recovery, the restoration of the heat content in the CML also shows a dependence on λ (Fig. 13). Increasing λ results in faster warming of the CML. This is more significant for the 2D simulations because the anomalous air–sea heat flux is the only way to restore the CML heat content; after 45 days, less than 20% of the CML heat content has been recovered when λ = 15 W m−2 K−1, whereas it is more than triple with a 10 time larger λ.

b. Sensitivity to the strength of wind stress

Winds play an important role in the upper-ocean dynamics because they can input mechanical energy and thereby generate strong turbulent mixing and destroy the stratification in the upper ocean. Besides, as demonstrated by Mahadevan et al. (2010), depending on their direction, winds may quicken or curtail the development of the submesoscale structures and thus hasten or delay the restratification process because submesoscale instability is crucial to the upper-ocean restratification because of its efficiency and rapidity.

Here, we only consider the changes in wind speed. A set of experiments were performed to study the influence of poststorm wind strength on the recovery of both the SST and the CML heat content. These experiments are same as the reference run but with different wind strength: the standard deviation of the zonal wind stress varies between 0.015 and 0.15 N m−2 in increments of 0.015 N m−2 ( is kept at half of as in Table 1; the magnitude of the resultant wind stress varies between 0.0135 and 0.135 N m−2 in increments of 0.0135 N m−2; 0.054 N m−2 is the reference run).

Opposite to the effect of changes in λ, an increase in wind stress can significantly delay the recovery of the SST (Fig. 14). This is because stronger winds enhance the vertical turbulent mixing, deepen the mixing layer, and thereby distribute heat over a greater depth, slowing down the warming of the surface water.6 The dependence of the SST recovery time on wind stress satisfies a power law with a scaling exponent of ~1.6 (or ~1 if wind power, which is defined as the wind speed cubed, is used). This means that changes in recovery time are steeper (faster) than those in wind strength. The intercept of the fitted power function is not zero, indicating that, even if there is no wind forcing, the SST still needs several days to recover. Indeed, an e-folding time of 5.7 days is obtained in an experiment without any wind forcing. This is because, besides the wind stress, the surface buoyancy loss at night can also lead to strong turbulent mixing via generation of penetrative convection, which mixes heat from the surface layer downward to the lower layer, resulting in retarded SST recovery (e.g., Kraus and Turner 1967).

Fig. 14.
Fig. 14.

The e-folding time (days) obtained using the exponential fit from the sensitivity experiments with different wind stress forcing. The solid curve is calculated using a power-law fit.

Citation: Journal of Physical Oceanography 42, 9; 10.1175/JPO-D-11-0209.1

As stated earlier, slow SSTA decay is accompanied by an increase in the efficiency of the upper-ocean heat uptake due to the dependence of the anomalous air–sea heat flux on the SSTA. The restoration of the CML heat content due to diabatic heating at day 45 more than doubles when the wind stress increases from 0.015 to 0.15 N m−2 (Fig. 15b). The relationship between (and correspondingly the magnitude of the wind stress) and the contribution of anomalous air–sea heat fluxes is approximately linear. However, the sensitivity of the contribution of baroclinic instability to the CML heat content is modest, varying between 26% and 30% with a general decrease as wind stress strengthens during the range of wind stress strength displayed (Fig. 15b). Because of this small canceling effect, the total restoration of the CML heat content increases at a little slower rate compared to the component attributable to the air–sea heat flux anomaly when increases up to a magnitude of 0.12 N m−2 (Fig. 15a). However, a plateau emerges as wind stress continues to increase, which is primarily due to enhanced vertical heat diffusion and entrainment of cooler water at the base of the CML.

Fig. 15.
Fig. 15.

(a) As in Fig. 14, but for the percent of heat content that has recovered in the CML after 45 days. (b) As in (a), but for the contribution from the base of the CML (dashed curve) and from the air–sea interface (solid curve).

Citation: Journal of Physical Oceanography 42, 9; 10.1175/JPO-D-11-0209.1

c. Sensitivity to TC-induced maximum mixing depth

For a fixed CML depth, changes in the TC-generated mixing depth, in response to TCs of different intensities or of different translation speeds, result in changes in the available potential energy and thus may affect the restratification process associated with baroclinic instability. To test the dependence on the mixing depth, we performed experiments varying hmax in Eq. (3) between 100 and 275 m in increments of 25 m (we will refer to this set of restratification experiments as intermediate-CML runs). The simple front adjustment model presented in section 2 suggests that the contribution of baroclinic instability to the CML heat budget is sensitive to the ratio between the TC-induced mixing depth and the CML depth (i.e., hmax/hcml) rather than the absolute value of the mixing depth hmax. Considering that hcml = 76.26 m, this corresponds to hmax/hcml in the range of 1.3–3.6. Note that the observations indicate that this ratio can be as large as 6 in extreme cases and for shallow prestorm mixed layers (e.g., Sakaida et al. 1998; Liu et al. 2007), indicating that our simulations are relevant for real cases. To further test this special dependence on hmax/hcml, we obtain two different climatological states with different CML depths by repeating presimulation, which is subject to wind stress of different strengths, σx = 0.03 and 0.12 N m−2 (σy is kept at half of σx as in Table 1), respectively. The respective CML depth obtained is 63.63 and 119.73 m (their subsequent restratification simulations are referred to as shallow-CML and deep-CML runs, respectively). Note that all the forcings are kept consistent between a presimulation and its restratification simulations.

The temporal evolutions of the domain-averaged SSTA in all the restratification simulations with different mixing depths are quite similar, despite the different magnitude of the initial SSTA (not shown), and accordingly there are no significant changes in the recovery time scale of the SST. This is consistent with the observational results suggesting that the SSTA disappears after approximately 1–2 weeks, independent of the magnitude of the surface cold anomaly and the storm intensity (Mei and Pasquero 2012, manuscript submitted to J. Climate).

The contribution of baroclinic instability to the recovery of the CML heat content in the intermediate-CML runs are shown as a function of hmax/hcml (asterisks in Fig. 16). Clearly, baroclinic instability contributes more as the normalized mixing depth increases. This is not surprising because a greater mixing depth produces a larger negative temperature anomaly in the CML, and as a result baroclinic adjustment induces larger vertical heat fluxes into the CML and thereby contributes more to the CML heat content recovery [note that the dependence of the contribution of baroclinic instability on hmax/hcml is stronger than the dependence of the CML heat content anomaly; see Eqs. (A3)(A5) for the simple model case]. A comparison with the simple model results indicates that the contribution of baroclinic instability is approximately 75% of that predicted by the simple model. This difference is mainly due to the fact that anomalous air–sea heat fluxes also contribute to the recovery of the CML heat content, reducing the contribution of baroclinic instability.

Fig. 16.
Fig. 16.

Contribution of baroclinic instability to the restoration of the CML heat content with various mixing depths in the intermediate-CML runs (asterisks), the shallow-CML runs (circles), and the deep-CML runs (dots). The results from the simple front adjustment model with same configurations are presented as the solid curve. The dashed curve is 0.75 times the simple model results.

Citation: Journal of Physical Oceanography 42, 9; 10.1175/JPO-D-11-0209.1

Figure 16 also indicates that simulations with significantly different hcml but similar hmax/hcml give similar results, confirming what is found using the simple front adjustment model. The lack of perfect superposition between results with different CML depth but same normalized maximum mixing layer depth is attributable to the different configuration of the simulations, because the wind stress forcing has to be varied to change the CML depth.

d. Sensitivity to the shape of the TC-induced mixing depth

TCs of different compactness and sizes have different radial wind structures and thereby may generate mixing depth with different across-TC-track shapes. In this section, we test whether the difference in the shape of the spatial variations of the mixing depth has impacts on the contribution of baroclinic instability to the CML heat content recovery.

In this series of experiments, only the value of k in Eq. (3) varies while the values of other parameters are the same as in the reference run. Figure 17a displays a few selected possible shapes of the TC-generated mixing depths (the thick curve corresponds to the setup in the reference run). The contribution of baroclinic instability is shown in Fig. 17b. It is evident that baroclinic instability contributes more when the mixing depth has a relatively constant slope, whereas the contribution is minimized when the mixing depth is set as a step function. The primitive equation ocean model results again are about 75% of those from the simple model (dashed curve in Fig. 17b). It is worth noting that the sign of the dependence may vary when other parameters change (e.g., hmax/hcml and n), as suggested by the results of the simple model reported in Fig. 4. Anyhow, the dependence on the value of k is relatively weak compared to the dependence on the mixing depth discussed above.

Fig. 17.
Fig. 17.

(a) Selected shapes of the spatial variation of the mixing depths used in the sensitivity tests. On the left side, from the bottom to the top, the values of k are 1 × 10−14, 2.5 × 10−5, 5 × 10−5, 1 × 10−4, 2 × 10−4, 4 × 10−4, 1 × 10−3, and 1 × 106. (b) Contribution of baroclinic instability to the recovery of the CML heat content as a function of kLy with Ly = 100 km. The solid curve is the results from the simple front adjustment model and the dashed curve is 0.75 times the simple model results.

Citation: Journal of Physical Oceanography 42, 9; 10.1175/JPO-D-11-0209.1

e. Sensitivity to addition of unperturbed water to the south of the model domain

In all the previous simulations, a wall boundary is employed at the southern edge of the front. In the real ocean, however, there is plenty of unperturbed warm water in the CML to the south of this edge. Thus, in this part we examine whether this artificial boundary may affect the evolution of the temperature field during the first 20 days when the contribution of baroclinic instability peaks and how the extension of the unperturbed water affects our conclusions. (As noted in footnote 1, the effect of northern wall boundary has been examined by using a trough instead of a front. We found no significant differences between using a trough and using a front.)

We carried out six additional experiments by adding water characterized with climatological temperatures (in the following we refer to it as unperturbed water) to the south of the domain of the reference run (i.e., y < 0 in Fig. 5) by factors of 0.2, 0.52, 1.0, 1.56, 2.2, and 3.0, respectively. These experiments are referred to as the 1.2-, 1.5-, 2.0-, 2.5-, 3.2-, and 4.0-size runs, respectively. The temporal evolutions of the cumulative contribution of baroclinic instability in the first three experiments are shown in Fig. 18a, together with the evolution from the reference run for comparison. There are no significant differences in the contribution of baroclinic instability in the runs with extended meridional domain from that in the reference run before day 20 when the contribution in the reference run peaks. This implies that the presence of the additional unperturbed water does not significantly influence the temperature evolution during the first 20 days. Examination of the SST evolution lends support to this implication (not shown).

Fig. 18.
Fig. 18.

(a) Temporal evolution of the cumulative contribution of baroclinic instability to the recovery of the CML heat content in the reference run (solid black), the 1.2-size run (solid gray), the 1.5-size run (dashed black), and the 2.0-size run (dashed gray). (b) Contribution of baroclinic instability to the recovery of the CML heat content as a function of the meridional size of the domain normalized by the meridional size in the reference run. The black curve shows the results from the simple front adjustment model, and the dashed curve is 0.75 times the simple model results.

Citation: Journal of Physical Oceanography 42, 9; 10.1175/JPO-D-11-0209.1

Baroclinic instability, however, continues to make a contribution to the CML heat content recovery in those extended-domain runs, whereas it ceases in the reference run. This dynamical process lasts longer when more unperturbed water is added to the south of the warm side of the front shown in Fig. 5. For example, the vertical dynamical fluxes associated with baroclinic adjustment approach zero around day 45 in the run with an extension of a factor of 0.52 (i.e., the 1.5-size run) while they disappear around day 75 in the 2.0-size run.

The total cumulative contribution of baroclinic instability in these extended-domain runs as well as in the reference run is given in Fig. 18b. This contribution increases quickly with increases in adding unperturbed water. Again, such a contribution is around 75% of that from the simple model. However, this positive tendency decelerates after the extension is comparable with the original meridional domain size; further extension results in a relatively small change in the contribution of baroclinic instability, which converges at about 50% of the total recovery of the CML heat content. This is because, in the run with a larger extension, baroclinic adjustment takes longer time, and as a result the contribution of anomalous air–sea heat fluxes becomes more important, which reduces the efficiency of the contribution of baroclinic instability measured as the ratio of the contribution in ROMS to that in the simple model.

Furthermore, baroclinic adjustment also leads to an uplift of the warm anomaly located in the climatological thermocline and a downward motion of the cold anomaly located above, as shown in Figs. 9b and 19. The uplift of the warm anomaly is greater when the unperturbed water is extended by a larger amount (Fig. 19a). For example, in the reference run the center of the warm anomaly is located at a depth of 145 m after baroclinic adjustment, whereas it is further lifted upward to a depth of 130 m in the 3.2-size run. This is because baroclinic instability operates to spread the anomalously warm water horizontally and this spreading is greater in the run with a larger domain, and as a result the warm anomaly is getting closer to the base of the CML. Such an uplift of the warm anomaly may have an important implication in term of its fate, which is important for the estimate of the long-term effect of TCs on the ocean heat content. In a recent study, Jansen et al. (2010) suggest that the dominant portion of the warm anomaly induced by the TC passage may be erased during the following winter because of a deepened mixed layer while assuming that the warm anomaly does not move vertically. If such an erasing process works in reality, then a lifted warm anomaly (owing to baroclinic instability) is more vulnerable and thus the net long-term effect of TCs on the upper-ocean heat content is even less important than suggested by Jansen et al. (2010).

Fig. 19.
Fig. 19.

(a) Temporal evolution of the depth at which the maximum value of the domain-averaged warm anomaly is located in the reference run (thick black), the 1.2-size run (thick gray), the 1.5-size run (thin black), and the 3.2-size run (thin gray). (b) As in (a), but for the domain-averaged cold anomaly.

Citation: Journal of Physical Oceanography 42, 9; 10.1175/JPO-D-11-0209.1

6. Summary and conclusions

This study has examined the restratification process of the upper ocean after the passage of a tropical cyclone (TC), with an emphasis on the role of baroclinic instability, by means of numerical simulations. A simple front adjustment model, excluding diabatic and diffusive processes, is first utilized to obtain an estimate of the maximum possible magnitude of the contribution of baroclinic instability to the recovery of the heat content in the climatological mixed layer (CML) and its dependencies. The model solutions suggest that the contribution of baroclinic instability increases with increases in the ratio of the mixing depth to the CML depth hmax/hcml and in the ratio of the areas of the unperturbed to perturbed water n. However, the contribution from baroclinic instability exhibits only a weak dependence on the spatial variation of the mixing depth k and is independent of the stratification below the CML.

A three-dimensional (3D) ocean model, ROMS, is then employed for process studies and to give a quantitative estimate of the respective contribution of baroclinic instability and anomalous air–sea heat fluxes to the restratification. Other processes that can contribute to the restratification, such as mesoscale eddies with their effects on lateral mixing, are not considered in this study. The model can well reproduce the satellite-observed e-folding time of the SSTA decay when it includes the baroclinic instability process and is initialized with moderate-amplitude eddy structures. The temporal evolution of temperature at various depths and of the stratification in the upper ocean in our idealized simulation are in good qualitative agreement with the measurements at the Bermuda Testbed Mooring site after the passage of Hurricane Fabian in 2003. The model ocean is well stratified after one month, with a net heat gain in the total water column that is considerably smaller than in absence of baroclinic instability. Diagnosis of the thermodynamic equation reveals that the upward heat flux at the base of the CML, due to the covariability of vertical velocity and temperature anomaly, dominates the heat budget of the CML during the first month. It is responsible for 2/3 of the warming in the first 3 weeks, demonstrating the importance of baroclinic instability in the upper-ocean restratification after the TC passage. This dynamical process is associated with the submesoscale cold filaments and rings with a spatial scale of a few kilometers.

A comparison between the 2D simulations that exclude baroclinic instability and the full 3D simulations further suggests that baroclinic instability significantly shortens the SST recovery time. The effects of baroclinic instability are twofold. First, baroclinic instability can input a considerable amount of heat into the CML and thereby enhance the warming rate of the surface water. Second, while it warms the CML, it cools the layer below and thus it is more efficient at restratifying the upper ocean compared to the anomalous air–sea heat flux. The established stratification in the CML can act as a barrier that inhibits the vertical turbulent mixing and downward heat diffusion. This further helps to accelerate the warming of the surface layer. Because the upward heat flux associated with baroclinic instability quickly reduces the SSTA, the corresponding anomalous air–sea heat flux and its contribution to the CML heat content recovery are relatively smaller in the 3D simulations compared to the 2D ones. The overall effect of including baroclinic instability, however, is to speed up the warming of the CML.

For both 2D and 3D simulations, increases in the surface temperature relaxation rate shorten the decay time of the SSTA and enhance the heat uptake efficiency of the upper ocean by anomalous air–sea heat fluxes. This results in a slight reduction in the contribution of baroclinic eddies to the recovery of the CML heat content in the 3D simulations. In contrast, as wind stress strengthens, the recovery of the SST slows down because of enhanced vertical turbulent mixing and the resultant increased downward heat diffusion. At the same time, the CML warms faster, as long as the wind stress is not too strong (the standard deviation of zonal wind stress is smaller than 0.12 N m−2). When wind stress further strengthens, more heat will be mixed downward underneath the CML instead of being used to heat the CML. The effect of changes in wind stress on the contribution of baroclinic instability, however, is modest.

Initializing the model with eddies of smaller magnitude slows down the SST recovery. However, this has a modest effect on the contribution of baroclinic instability and thus does not alter our main conclusions.

The dependence of the contribution of baroclinic instability on hmax/hcml, n, and k is also examined in ROMS. The results are quantitatively consistent with those obtained from the simple front adjustment model and are approximately 75% of the latter. The differences are mainly attributed to the fact that in ROMS anomalous air–sea heat fluxes also make a contribution and thus suppress that of baroclinic instability.

Based on these calculations, we expect that, in the real ocean with a typical CML depth of 50 m, when the maximum mixing depth generated by the TC passage reaches 150 m, baroclinic instability may contribute approximately 50% of the restoration of the CML heat content during the restratification process. Those results have important implications in terms of estimating the long-term effect of TCs on the upper-ocean heat content. In previous studies quantifying the role of TCs in pumping heat into the ocean, a basic assumption made is that the cold anomaly in the CML is totally removed by anomalous air–sea heat fluxes while the warm anomaly in the climatological thermocline persists, resulting in a net warming in the entire water column (e.g., Emanuel 2001; Sriver and Huber 2007). However, our results based on idealized simulations suggest that baroclinic instability can make a substantial contribution to the heat budget of the CML after the TC passage through returning about half of the warming back into the CML. Therefore, previous work may have significantly overestimated the role of TCs played in the ocean heat budget. Although this study contributes to putting an upper estimate on the effect of TCs on the ocean heat content, it still remains to be estimated their precise long-term effect on the upper-ocean state, which probably can be obtained only by further observational work.

Acknowledgments

This work was supported by NASA Headquarters under the NASA Earth and Space Science Fellowship Program Grant NNX10AP30H. We thank Prof. Francois Primeau and Prof. James McWilliams for helpful discussions and valuable comments on an earlier version of this manuscript. Thanks also to the anonymous reviewers for their comments, which helped us to improve the manuscript. We thank Prof. Todd Dupont and Prof. Keith Moore for sharing computing resources. We also thank Prof. Kerry Emanuel for sharing the compiled tropical cyclone best-track data with modified wind speeds and Prof. Tommy Dickey for the BTM measurements so we can reproduce Figs. 8d, 9a, and 10a of Black and Dickey (2008). TMI data are produced by Remote Sensing Systems and sponsored by the NASA Earth Science MEaSUREs DISCOVER Project. Data are available online (at http://www.remss.com).

APPENDIX A

Analytical Solution to the Simple Front Adjustment Model

We start with a simpler situation with n = 0 in Eq. (3); then, the mixing depth h as a function of y in the case with kLy→+∞ m is given by a step function,
ea1
The corresponding SSTA as a function of y is
ea2
Then, the decrease in the CML heat content due to the downward heat transport can be written as
ea3
After the adjustment, water within the climatological thermocline with a temperature greater than T0 + ΔSST should rise and overly the water that is initially located in the cold half of the CML (i.e., Ly/2 < yLy). The thickness of this thermocline water is Δh = |ΔSST/Γ| = (hmaxhcml)2/(2hmax). Then there are two possibilities, depending on the ratio of Δh and hcml. If Δh < hcml [i.e., (hmaxhcml)2/(2hmax) < hcml or ], then only part of the water located in the CML of the cold wake is replaced by the thermocline water. Accordingly, the CML heat content change owing to baroclinic adjustment is
ea4
The corresponding contribution of baroclinic instability to the recovery of the CML heat content is
ea5
On the other hand, if Δhhcml [i.e., ], then the water located in the CML of the cold wake is completely replaced by the thermocline water. The corresponding CML heat content change attributable to baroclinic adjustment is
ea6
Then, the contribution of baroclinic instability to the recovery of the CML heat content is
ea7
We may write Eqs. (A6) and (A7) together as
ea8
In the real ocean, the warmer side of the domain may extend far away, which may affect the value of r. Thus, we extend the region to the left of y = 0 in Fig. 2a by n times Ly with a new meridional size of (n + 1)Ly. Then, the corresponding contribution of baroclinic instability r is
ea9
The above equation can be further written as
ea10
Clearly, the value of r depends on the relative deepening of mixing layer (i.e., hmax/hcml) instead of the absolute deepening hmax and also depends on the relative size on the unperturbed side of the domain in y direction (i.e., n). Interestingly, it has no dependence on the stratification in the thermocline Γ.

Finding the solutions to the case in Fig. 2b in an analytical way is hard and to the case in Fig. 2c is not that straightforward. The analytical solutions to the case in Fig. 2c can be obtained only when the values of hcml, Ly, n, and the slope of the mixing depth are specified. The detailed derivatives are not given here.

APPENDIX B

Generation of Initial Disturbance

To initiate the development of baroclinic instability along the front, the initial temperature field is usually seeded with small random white noise in nonlinear numerical simulations (e.g., Boccaletti et al. 2007). In reality, however, finite-amplitude eddies are rarely observed to proceed from small disturbances of a nearly unperturbed basic state. Also, in the model the time needed for a small perturbation to reach finite amplitude on a realistic thermal front is much longer than the observed time scale for the evolution of eddies in the ocean. For example, in the modeling study by Boccaletti et al. (2007), the disturbances reach their nonlinear stage after 12 days of integration. In contrast, D’Asaro (2003) observed a well-defined eddy in the cold wake of Hurricane Dennis (1999) 3 days after its passage (see his Fig. 11). A similar situation is found in the atmospheric context. As noted by numerous observational, theoretical, and modeling studies in synoptic meteorology, initial conditions of preexisting finite-amplitude disturbances in the upper layer are conducive to or necessary for the process of (type B) cyclogenesis (e.g., Sanders 1988). When those disturbances approach the regions with strong surface baroclinicity, they grow rapidly and initiate surface cyclogenesis in a short time period by tapping the available potential energy stored in the low troposphere (e.g., Farrell 1985). Accordingly, initial disturbances like wave packets have been employed to study the growth of baroclinic waves in the atmosphere (e.g., Harnik and Chang 2004). Following this approach, in the restratification simulations of this study the model temperature field is seeded with eddy-like structures or a random red noise of relatively large amplitudes. This is also supported by observations that the temperature field has a red spatial spectrum and that mesoscale structures are ubiquitous in the ocean (e.g., Isern-Fontanet et al. 2006, 2008).

The eddy-like structures are obtained from an unforced experiment. In this experiment, the model is initialized with the temperature front described in section 3, on which infinitesimal white noise O(10−5°C) is superposed. Then the front evolves freely with neither wind nor heat flux forcing. It takes about 17 days for the eddy field to reach its nonlinear stage. After removing the zonal mean, the obtained 3D eddy temperature field from different days during the linear stage is superposed on the initial frontal structure shown in Fig. 5 to initiate the restratification simulations in sections 4 and 5. Note that the disturbance is added to the temperature field only and not to the velocity field. For this reason, the clusters of correlated temperature are not real coherent structures. In section 5a, the eddy temperature fields are taken from the output of the unforced experiment on days 2, 6, 10, 13, 15 and 16 and the corresponding simulations are denoted as ExpSets 1–6, respectively. The magnitudes of the eddy field are 10−4°, 10−3°, 10−2°, 0.04°, 0.14°, and 0.22°C, respectively. The reference run corresponds to the ExpSet 5 run with the Newtonian cooling rate λ = 60 W m−2 K−1.

For comparison, we also initialized the model with a vertically uniform horizontal red noise field with an amplitude of 0.15°C and the 2D isotropic energy-spectrum function E(k) ∝ k−2 (k is the wavenumber). The obtained SSTA evolution is almost identical to that obtained from the reference run, except that in the red-noise experiment it takes a couple of days for the noise to develop an organized structure (not shown). This suggests that an experiment initialized with any relatively large-amplitude random thermal field with an appropriate red energy spectrum can produce similar results. For convenience, we use the eddy-like structures stated above to initialize the model and test the sensitivity of the model results to the amplitude of the initial disturbance.

APPENDIX C

Observational Data and Methods for the Composite Analysis of the SST Response to Tropical Cyclones

Daily SST data from the version-4 Tropical Rain Measuring Mission (TRMM) Microwave Imager (TMI; Wentz et al. 2000) and TC-track data available at 6-h intervals from the National Hurricane Center best-track dataset (McAdie et al. 2009) and the Joint Typhoon Warning Center best-track dataset (Chu et al. 2002) over the period between 1997 and 2009 are used to examine the SST response to the passage of TCs. For each storm location, a 5° × 5° box is defined around the storm center. Holding the location of the box fixed, the mean temperature anomaly 〈T′〉 is calculated from the approximately 400 SST grid points in the box (here, an angular bracket indicates a spatial average over the box, and a prime refers to the temporal anomaly with respect to the sum of the climatological seasonal cycle and the linear trend over the whole length of the time series). Then the evolution of these mean values are calculated starting from 2 months before the TC passage until 4 months after it, for a total of 181 daily temperature data for each box. Finally, the obtained evolutions of SSTA for each box are used to perform a composite analysis based on storm intensity. Because the storm-induced mixing layer is relatively deep (200 m) in the reference run of our simulations and it is assumed that the heat content is conserved over the water column during the TC passage, which is more relevant to the passage of a strong TC, we validate the modeled SST evolution using the observations associated with the passage of TCs of hurricane intensity (i.e., the dashed curve in Fig. 6). More details on the observational data analysis can be found in Mei and Pasquero (2012, manuscript submitted to J. Climate).

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1

It is numerically more efficient to study the evolution of one front rather than of the whole wake that can be described as comprised by two fronts. We performed a simulation using a trough with doubled size in the meridional direction and compared the obtained evolutions of SST and heat content with those from the reference run in section 4. No significant differences were found. This indicates that a wall boundary at the northern edge (i.e., y = Ly = 100 km in Fig. 5) does not affect our conclusions.

2

The diurnal evolution of incoming solar radiation Qsw can be expressed as (Stull 1988) Qsw = STk sinψ for daytime and Qsw = 0 for nighttime (S is the solar irradiance, Tk is the net sky transmissivity, and ψ is the solar zenith angle). For a certain latitude and a certain time period of a year, sinψ ∝ cos(2πt/24 h) (t is the local time). Thus, without considering the diurnal variations of cloud cover, the solar radiation Qsw ∝ cos(2πt/24 h). Such an expression has been confirmed by observations in Bacellar et al. (2009), where it is shown that the incoming solar radiation at the surface can be fitted using a cosine function (see their Fig. 10). Similar observational results can also be found in Mlynczak et al. (2006) (see their Fig. 11).

3

This is a crude representation of sea surface heat loss. Usually the net longwave radiation and turbulent fluxes are calculated using bulk formulas in models. However, the lack of high-resolution observations of variables such as humidity, wind speed, and cloud cover does not allow us to calculate those heat fluxes in typical poststorm conditions. This together with the idealized nature of our simulation is the primary reason for the use of a restoration term to represent surface heat loss. Actually, this restoration term is widely used in climate modeling studies. For the present study, this approximation is reasonable, because it was assumed that winds are close to their climatology after the TC passage and an appropriate value of λ was chosen to represent such a normal condition. The value of λ given in Table 1 is chosen based on the space-dependence argument by Bretherton (1982) and on the estimation by PMN08 (see their Fig. 16). Note that the sensitivity of the results to varying values of λ is described in section 5a.

4

Here, the mixed layer depth is defined as the depth where the temperature is 0.2°C cooler than SST (e.g., de Boyer Montegut et al. 2004), and the criterion for identifying a quasi-steady state is that the deepening rate of mixed layer is less than 0.1 m month−1.

5

The 2D simulations were achieved by allowing only one grid box in the zonal direction, resulting in the removal of the zonally asymmetric baroclinic instability, whereas the setup in the meridional and vertical directions is the same as the full 3D simulations.

6

Another effect of strengthening the poststorm wind stress, which is not included in our model, is to enhance both evaporation and sensible heat and thus extract more heat from the surface water. This can also delay the recovery of the SST. The opposite situation applies to weakening the poststorm wind stress. Accordingly, we expect that the range of the e-folding time shown in Fig. 14 will be larger when such a dependence of surface heat loss on wind speed is included.

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  • Alexander, M. A., and C. Deser, 1995: A mechanism for the recurrence of wintertime midlatitude SST anomalies. J. Phys. Oceanogr., 25, 122137.

    • Search Google Scholar
    • Export Citation
  • Alexander, M. A., and C. Penland, 1996: Variability in a mixed layer ocean model driven by stochastic atmospheric forcing. J. Climate, 9, 24242442.

    • Search Google Scholar
    • Export Citation
  • Allen, J. T., and D. A. Smeed, 1996: Potential vorticity and vertical velocity at the Iceland-Faeroes Front. J. Phys. Oceanogr., 26, 26112634.

    • Search Google Scholar
    • Export Citation
  • Bacellar, S., A. P. Oliveira, J. Soares, and J. Servain, 2009: Assessing the diurnal evolution of surface radiation balance over the western region of tropical Atlantic Ocean using in situ measurements carried out during the FluTuA project. Meteor. Appl., 16, 255266.

    • Search Google Scholar
    • Export Citation
  • Black, P. G., and Coauthors, 2007: Air–sea exchange in hurricanes: Synthesis of observations from the Coupled Boundary Layer Air–Sea Transfer experiment. Bull. Amer. Meteor. Soc., 88, 357374.

    • Search Google Scholar
    • Export Citation
  • Black, W. J., and T. D. Dickey, 2008: Observations and analysis of upper ocean responses to tropical storms and hurricanes in the vicinity of Bermuda. J. Geophys. Res., 113, C08009, doi:10.1029/2007JC004358.

    • Search Google Scholar
    • Export Citation
  • Boccaletti, G., R. Ferrari, and B. Fox-Kemper, 2007: Mixed layer instabilities and restratification. J. Phys. Oceanogr., 37, 22282250.

    • Search Google Scholar
    • Export Citation
  • Bretherton, F. P., 1982: Ocean climate modeling. Prog. Oceanogr., 11, 93129.

  • Brooks, D. A., 1983: The wake of Hurricane Allen in the western Gulf of Mexico. J. Phys. Oceanogr., 13, 117129.