Polar lows are intense mesoscale cyclones with surface winds often exceeding 25 m s−1. Most frequently they are observed over the Nordic Seas during the winter half-year (November–April). They affect both coastal communities and seafarers, sometimes with the loss of lives. Polar low formation is often triggered by an outbreak of cold and dry air from the ice-covered Arctic over the relatively warm Nordic Seas. Their small size and relatively rapid development, together with a sparse observational network, make them difficult to forecast (Rasmussen and Turner, 2003).
The study of polar lows has included considerable controversy. There has been a range of different definitions and different names. It has been especially difficult to agree about the forcing mechanism. Today it is often accepted that polar lows may may be forced by different mechanisms. The book edited by Rasmussen and Turner (2003) is often regarded as the standard literature on the subject. It gives the following definition:
A polar low is a small but fairly intense maritime cyclone that forms poleward of the main baroclinic zone (the polar front or other major baroclinic zone). The horizontal scale of the polar low is approximately between 200 and 1000 kilometres and surface winds near or above gale force.
Note that no reference to any forcing mechanism is given. Rasmussen and Turner (2003) suggest that their definition can be extended by specifying the mechanism responsible for the development.
Several investigators have emphasized the baroclinic structure of polar lows. Harrold and Browning (1969) and Mansfield (1974) find that baroclinic instability theory is consistent with observations of the wavelengths and phase speeds of the disturbances. However, this is based on the assumption of shallow normal-mode growth. Reed (1979) suggested that this assumption of normal-mode growth within a shallow layer conflicted with the experience of forecasters, who found that upper-level vorticity maxima are associated with polar lows. Reed and Duncan (1987) find partial agreement between observations and a linear quasi-geostrophic dry baroclinic model. However, they argue that the development must have been influenced by some other mechanism, which they believe is organized deep convection.
For as long as polar lows have been studied, researchers have noted their similarities with tropical cyclones. The similarities include exclusive formation over the ocean, rapid decay over land, strong deep convection, spiral cloud structures and often a clear eye. Unsurprisingly, theories for describing polar lows have included theories originally developed for describing tropical cyclones, such as Conditional Instability of a Second Kind (CISK), formulated by Charney and Eliassen (1964), and Air–Sea Interaction Instability (ASII), formulated by Emanuel (1986a) and later renamed Wind-Induced Surface Heat Exchange (WISHE).
These theories assume that (as with tropical cyclones) some other mechanism not dependent on organized deep convection is responsible for the creation of the initial disturbance. This process can be baroclinic instability. Many researchers describe polar lows as belonging to a spectrum ranging from predominantly baroclinic to predominantly convective. Over the lifetime of a polar low, its position in this spectrum may change.
Saetra et al. (2008) show that strong wind over the Nordic Seas, e.g. associated with polar lows, can entrain warm subsurface water into the surface layer and thus cause the sea-surface temperature (SST) to rise. (A situation with warmer water below colder does not in this case imply that the stratification is unstable, since the warmer water has a higher density caused by its high salinity.) They present observational evidence showing an example of an SST increase by up to 2 °C coinciding with the passage of a polar low. The example is from an area where the warm North Atlantic Current is especially pronounced due to topographical steering. The effect on the SST would be the same if the wind were caused by a synoptic-scale low-pressure system, but since such a system is not immediately influenced by the SST there would also be little feedback on the low. Polar lows, however, are often assumed to be dependent on the heat fluxes from the ocean. WISHE assumes this explicitly and the exclusive formation of polar lows over the ocean is a strong indication supporting this hypothesis. It is therefore possible that an SST increase caused by a polar low would feed back on the low. The question of such feedback is raised by Saetra et al. (2008). It is even likelier that a polar low will be exposed to a changing SST due to its horizontal translation. The SST difference covered during one day by a typical polar-low track can often be around 3 °C.
The effect of the upper ocean on the intensity of tropical cyclones has been investigated by several researchers, e.g. Bender et al. (1993), Shade and Emanuel (1999), Sutyrin and Khain (1984). The SST always falls during the passage of a tropical cyclone. This is because at low latitudes the deeper water entrained into the ocean surface layer is always colder. The cooling of the ocean surface inhibits the intensification of tropical cyclones, particularly those that are slow-moving. This inhibiting effect is reduced in regions with a deeper ocean surface layer, where there is greater insulation of the surface from colder water under the thermocline. For example, the effect of warm core rings or the Loop Current in the Gulf of Mexico have been studied (Hong et al., 2000; Shay et al., 2000). A parallel to this in the Nordic Sea environment could be a polar low passing over a subsurface warm core and possibly being intensified by the rising SST. However, the effect of the ocean on the intensity of polar lows has hitherto not been systematically investigated. Emanuel (1986b) (E86b hereafter), using the WISHE theory, makes a brief theoretical investigation of how the maximum intensity of a polar low is influenced by the SST. He finds that if the SST increases by 1 °C the theoretical maximum sea-level pressure (SLP) depression deepens by around 7 hPa.
In the present investigation we study the effect of an SST change on the intensity of polar lows simulated in a numerical model. We use an ensemble approach where several control runs with constant SSTs are compared with runs with perturbed SSTs. The intensity is measured in terms of SLP depression and maximum azimuthal wind. The SST perturbations are prescribed and may be due to horizontal translation of the low, vertical mixing in the upper ocean or any other process.
In an additional section we try to shed some light on the rather surprising result from the numerical simulations. We do this by relating our results to the theoretical investigation of E86b. In his investigation Emanuel assumes that the polar low works as an Carnot engine, where the energy input to the low comes from air being heated at high temperature in the boundary layer and cooled at low temperature higher up in the atmosphere. He calculates the maximum possible heating for three different SSTs by assuming that the air reaches the SST and is saturated as it leaves the boundary layer. He further estimates the cooling temperature as a value rather close to the temperature of the tropopause. He finds that the maximum intensity is rather sensitive to the SST, but also that the observed polar low from his example is far from this maximum. In what we believe is a novel and qualitatively useful approach, we develop an analytical method for estimating how sensitive the actual (rather than the maximum) input from the Carnot engine is to changes in the SST.
In section 2 we describe the numerical model and how we use it. In section 3 we present the results of the simulations. In section 4 we suggest how the result should be explained in the framework of the WISHE theory. Finally a discussion and a summary are given in section 5.
The numerical model used in this study is a non-hydrostatic axisymmetric model that was originally developed to simulate hurricanes by Rotunno and Emanuel (1987) and adapted to simulate polar lows by Emanuel and Rotunno (1989). This model was chosen since it allows for an assessment of the intensification of a cyclone purely through air–sea interaction and is relatively quick to run compared with three-dimensional models. It has a long pedigree of use in many studies of hurricanes, too numerous to list here, and has also been used to simulate polar lows by Craig (1995) and Craig and Gray (1996). Convection is represented explicitly and prognostic equations for momentum, potential temperature, pressure, water vapour and ice are used. The turbulence parametrization is based on a first-order Richardson-number-dependent eddy viscosity. The model version used in this study is the same as that used by Craig (1995), who modified the model of Emanuel and Rotunno (1989) by changing the boundary conditions and introducing different parametrization of the long-wave radiation and microphysics schemes.
The model setup is the same as that used by Gray and Craig (1998). Key model parameters are summarized in Table I. The large domain size (radius of 2500 km and height of 18 km) was shown by Craig and Gray (1996) to remove a dependence of the intensification rate on domain size that had been found for smaller domains. Sponge layers at the upper and outer boundaries are included to absorb gravity waves. The initial vertical temperature and humidity profiles at the centre of the vortex were based on radiosonde data from 1200 UTC on 13 December 1982 over Bear Island in the Norwegian Sea, as used by Emanuel and Rotunno (1989). We use their ‘warm, moist’ profile; see Figure 1(a). The initial wind field is a baroclinic warm-core vortex as described by Craig (1995), depicted in Figure 1(b). The maximum near-surface wind speed is set to 10 m s−1 at a radius of 50 km. The wind speed decreases linearly with height to zero at the tropopause (6.5 km). In order to satisfy suitable boundary conditions, there is a weak anticyclonic vortex aloft, which vanishes at the base of the upper sponge layer at 13.5 km (see Emanuel and Rotunno, 1989 for details). The model is run with a constant Coriolis parameter calculated for 70°N (1.36 × 10−4 s−1).
Table I. Model configuration for all simulations.
Horizontal grid size
Vertical grid size
Height of domain
Radius of domain
Thickness of upper sponge layer
Thickness of outer sponge layer
To account for random variability and the sensitivity of nonlinear systems to small perturbations, we use ensembles of model simulations to study the SST response. First, five control simulations with fixed SST during the whole integration period are run. The control simulations are run with SST = 277, 278, 279, 280 and 281 K, respectively. Then, perturbations of these control runs are made by introducing an instantaneous SST change at one stage during each of the simulations. These SST changes are introduced after 20, 40, 60, 80 or 100 h of simulation. At each of these times, four different SST perturbations are made. The perturbations are from the actual control-run SST to the SST of one of the other control simulations. As an example, at 20 h simulation, the control simulation with SST = 278 K is perturbed to 277, 279, 280 and 281 K. Accordingly, each time of perturbation produces an ensemble of 20 perturbed members, four from each of the five control simulations. The SST response is estimated as the mean over each ensemble.
To get values for how sensitive the cyclones are to SST changes, we subtract the SLP depression and the maximum azimuthal wind of the control simulations from the perturbed simulations. The SST sensitivity of the SLP depression, SPSST, and the sensitivity of the maximum azimuthal wind, SWSST, are defined as
where δ(Δp) is the difference in cyclone SLP depression, δ(Vmax) is the difference in maximum azimuthal wind and δTss is the SST perturbation. As an example, we look at the 10 h average of the SLP depression from radius 400 km to the centre, given in hPa. We take the result from the simulation where the initial SST at 278 K after 60 h is perturbed to 280 K. From that result we subtract the result of the control simulation with a constant SST at 278 K and divide by the SST perturbation, in this case +2 K. We obtain a time series of SPSST with the unit of hPa K−1. The time series has no values until the time of the perturbation at 60 h. For SWSST the unit is m s−1 K−1.
We arrange the result from each simulation in 10 h bins and average over each bin, i.e. each simulation produces time series for the times 5 h, 15 h, 25 h, ….
Figure 2 shows the mean velocities for the 10 h bin centred around 115 h for the control simulation with a constant SST at 279 K. At this period the simulated polar low has reached a state of relatively constant intensity, the quasi-steady phase. The main zone of convergence (negative radial velocity in the top panel) is confined to a region from 80 to 400 km radius. We identify the inner and the outer extent of this zone as the eyewall and the cyclone periphery, respectively. The convergence zone is tilting downwards and intensifying towards the centre, with the highest magnitude at 150 km radius. The mean height of zero radial velocity in this zone is around 3 km. Above the convergence layer is the outflow layer with radial velocities of similar magnitude, but with maximum intensity at 300 km radius. A branch of the outflow layer extends to a large radius at a height of 7 km (at the tropopause).
The middle panel shows a strong cyclone with maximum azimuthal wind at around 100 km in radius. Here the cyclonic velocities reach more than 8 km in height. The eye region is clearly seen as a zone of very small wind speeds that occupies the inner 30 km of the polar low. Typically for polar lows, this is considerably larger than the eyes of observed and simulated tropical cyclones. Outside the cyclone there is a high-altitude anticyclone, caused by the conservation of angular momentum in the outflow. The wind speed in the anticyclone is about two thirds of the wind speed in the cyclone.
The vertical velocity (bottom panel) varies greatly in both time and space, with short ( 1 h) convective bursts occurring at all radii outside the eye. The eyewall, at 80 km, is the only region of constant convection and here strong positive vertical velocities reach all the way to 7 km height. If the velocities are averaged over 10 h, as in Figure 2, the variations are smoothed and a wider zone of mean upward motion outside the eyewall is revealed. What we see is that the upward motion of air parcels out of the boundary layer, either in convective bursts or in the eyewall, is relatively fast. The downward motion on the other hand is generally a very slow process.
Figure 3 shows the mean equivalent potential temperature and temperature, the humidity and the SLP for the same period and simulation as in Figure 2. Both the isotherms for the temperature (top panel) and the isolines for humidity (middle panel) are tilting downwards from the eyewall and outwards, corresponding to both sensible and latent heating as the air converges towards the main convection zone. At 8 km height we find a local temperature minimum. This is caused by the strong convection lifting the air above its level of neutral buoyancy and quite possibly also by strong radiative cooling on top of the cloud layer.
In the convergence layer, the equivalent potential temperature (top panel) increases towards the eyewall. Close to the eyewall the isotherms are nearly vertical up to around 6 km. At larger radii the near-vertical isotherms of equivalent potential temperature are restricted to the lowest 3 km of the model domain. This is a key indicator of the vertical extent of the well-mixed boundary layer. Collocated with the local temperature minimum at 8 km height, the isotherms for equivalent potential temperature are elevated compared with their level at 600 km radius, once again showing the effects of strong convection and strong radiative cooling.
The SLP (bottom panel) is lowest and uniform across the eye region. From the eyewall outwards the SLP increases with radius, with the largest gradient at the eyewall. The drop of the SLP gradient at 600 km radius is one of several local minima. At larger radii (not shown) the gradient pattern is similar to that between 300 and 600 km.
Figure 4 shows the development of the cyclone (400 km) SLP depression and maximum azimuthal wind for the control simulations with constant SSTs. After 20 h the cyclones start to intensify and after around 80 h the intensity is relatively constant. We will refer hereafter to the intensifying phase and the quasi-steady phase, respectively. As might be expected, the simulations with higher SST develop deeper SLP depressions. The difference between the control simulations gives a first indication of the magnitude of how sensitive (the simulated) polar lows are to SST changes. The SLP depression difference is of the order of 1 hPa K−1, but with great variation. For the maximum azimuthal wind the picture is similar, with a difference of the order of 1 m s−1 K−1, but with even weaker correlation to the SST. The wind also levels off earlier than the SLP depression.
We note immediately that the impact of the SST is smaller than we had anticipated. Differences as small as 1 hPa or 1 m s−1 will be difficult to measure in real polar lows, or to distinguish from internal variability.
To obtain a more certain value for SST sensitivity we turn to the perturbed simulations. Figure 5 shows the ensemble means for SPSST (top panel) and SWSST (bottom panel). Note that the horizontal axes show the time from the SST changes. As we expect, the sensitivity of the SLP depression is negative (an SST increase leads to a deeper SLP depression) and the sensitivity of the maximum azimuthal wind is positive (an SST increase leads to a higher maximum azimuthal wind).
In addition to the development of a cyclone SLP depression in the centre of the model domain, all of the simulations show a domain-wide drift towards increasing pressure. This tendency increases with higher SST, which to some degree confuses our sensitivity study. We chose to focus on the SLP depression from 400 km radius because it shows a quasi-steady phase most similar to the maximum wind. We believe this is as close to an objective choice we can get.
For SPSST a pattern emerges. First there is a transition period of about 20 h when the SPSST values increase. After the transition period, the model relatively consistently gives SPSST values between −0.5 and −1.0 hPa K−1. For SWSST the picture is once again similar but less clear. Generally the response is less than 1 m s−1 K−1. We note that there appears to be no systematic difference in the response depending on how far into the simulation the SST change takes place. Specifically, there is no apparent difference between the intensifying phase and the quasi-steady phase.
We also note that the transition period in Figure 5 is of similar duration to the time before the intensifying phase at the very beginning of the simulations in Figure 4. This response time can be compared with an estimate of the time an average air parcel spends in the boundary layer. We assume that the inner and outer limits of the cyclone boundary layer are at radii of 80 and 400 km, respectively, and that the radial velocity in the boundary layer is U = −3 m s−1. Then the residence time is 30 h. We find it intuitively plausible that the response time of a polar low should be similar to the duration of the changed forcing on the circulating air.
We want to know whether the response of the numerical model is proportional to the SST perturbations. To test this, the simulations are arranged into ensembles according to the size of the perturbations. The mean SPSST and the mean SWSST of these ensembles are shown in Table II, second and third columns, respectively. The variation between the ensemble means around the total mean is less than 15% and there is no trend. This indicates that the cyclone intensity changes are on average proportional to the size of the SST perturbations, both in terms of maximum azimuthal wind and SLP depression.
Table II. Sensitivity of the perturbed simulations. The values are averaged over the period 0–140 h after the SST perturbations. Data from the intensifying phase (less than 80 h after the start of the simulations) are excluded.
(m s−1 K−1)
The bottom row of Table II shows the mean sensitivity for all perturbed simulations. The mean value for SWSST is +0.60 m s−1 K−1 and for SPSST it is −0.57 hPa K−1.
4. Why the sensitivity is so limited
4.1. Emanuel's maximum-intensity calculations
The sensitivity presented in section 3 surprises us as low. Such a limited sensitivity will be difficult to observe in real polar lows and difficult to distinguish from internal variability or changes in other forcing.
The sensitivity presented by E86b is an order of magnitude larger, although it should immediately be pointed out that he studies the theoretical maximum intensity. Here we will use the WISHE theory to see if the result of E86b can be reconciled with that of the present investigation.
According to WISHE, the polar low works as a Carnot engine. The input of internal energy, E, comes from heat fluxes from the ocean, with a subsequent cooling higher up in the atmosphere. A schematic illustration of the Carnot circuit is shown in Figure 6. E can be calculated by
where cp is the heat capacity at constant pressure, Tb and Tout are the mean temperatures during the heating and cooling phases of the Carnot circuit, respectively, and θec and θe0 are the equivalent potential temperatures at radii rc and r0, respectively; see Figure 6. Equation (3) is derived in Appendix A.
We note that the mean temperatures Tb and Tout are not the arithmetic mean temperatures of their entire respective parts of the Carnot circuit, but the integrated mean at which the change of equivalent potential temperature takes place. Tb can be approximated by the arithmetic mean of the temperatures at radii rc and r0. E86b assumes that the change from θec to θe0 takes place where the moist adiabats of the boundary layer intersects a sounding at large radius from the cyclone centre. This puts the cooling temperature, Tout, rather close to the temperature of the tropopause. E86b finds the maximum E by calculating θec from saturation at the SST.
The input of internal energy is balanced by work. In the WISHE theory, this work consists of work against friction in the boundary layer, Wb, and work by the ambient atmosphere to restore the angular momentum, Wout:
Note that Wout is a source and has the opposite sign from Wb. Now, the Bernoulli equation states that along a streamline there is a balance between friction, kinetic energy gradient and pressure gradient. Wb can be found by integrating the Bernoulli equation along an average streamline in the boundary layer (from radius r0 to radius rc in Figure 6):
where Δp is the pressure difference towards the centre and Δ(V2)/2 is the increase of kinetic energy towards the centre. Inserting Eq. (5) into Eq. (4) gives the full expression for the internal energy input:
E86b argues that Wout is small and can be ignored. He also ignores the change of kinetic energy, so that the internal energy input is balanced only by the depression term. This yields
His calculations, based on observations of a polar low investigated by Rasmussen (1985), can be summarized as
The ρ value given above is not explicitly stated in E86b, but it is the value apparently used. The calculated SLP depression is in good agreement with what can be estimated from the observations (Δp < −13 hPa). We should note that the sounding used by Emanuel to estimate ambient conditions outside the polar low is the same as the sounding we use to initialize the numerical model. This is probably why we get a very similar result if we perform the corresponding calculations using values from our numerical simulations, for example from Figure 3. We estimate near-surface values by assuming that the potential temperature and the humidity are the same as in the lowest model level. The difference from E86b is that in our simulation the air is heated to a higher equivalent potential temperature (θec = 289 K), which yields a deeper calculated SLP depression (Δp = −25 hPa), This calculated SLP depression is also in good agreement with the simulated SLP depression.
E86b continues with finding the theoretical maximum SLP depression by calculating θec for different SSTs. All other parameters (except the central SLP) are kept constant and the air is assumed to have reached SST and to be saturated as it ascends out of the boundary layer. This yields
(E86b approximates the temperature at lifted condensation level with a constant, which appears to be 261 K.) When we perform the same calculations using the values from Figure 3 we get almost the same result. E86b notes that ‘Evidently, a very strong polar low could have developed if the storm had remained over warm water long enough and if the boundary layer physics had permitted surface air near the core to approach saturation at sea water temperature’. In section 3 we found that our numerical model did reach quasi-steady states, but not with intensities resembling the examples above. We conclude that the boundary-layer physics is a limiting factor for the intensity.
The sensitivity of the theoretical maximum SLP depression calculated according to E86b is around −7 hPa K−1, or an order of magnitude higher than the values for SPSST found in section 3. We identify two fundamental features of the E86b calculations likely to be responsible for this discrepancy.
(1)The θec change imposed by the SST change is the same as the change of the theoretical upper bound for θec.
(2)The kinetic terms are ignored. This refers to both the ambient atmosphere input of angular momentum, Wout, in Eq. (4) and the change of kinetic energy in the Bernoulli equation (5).
If θec is far from its upper bound before an SST change, it seems reasonable that the impact of the SST change also is far from the change of the upper bound. In subsection 4.2 we try to estimate the impact by relating the SST and the equivalent temperature in the boundary layer to the sea-surface heat fluxes. In subsection 4.3 we estimate the magnitude of the kinetic terms and the effect of including them.
4.2. The impact of sea-surface heat fluxes
We can greatly simplify the estimation of the impact of heat fluxes by treating the sensible and the latent heat together† in the common parameter equivalent temperature, Te, defined in Appendix A.
The equivalent temperature of the air ascending out of the boundary layer, Tec (at radius rc in Figure 6), can be calculated as
where Te0 is the equivalent temperature at radius r0, the second term on the right-hand side is the impact of the sea-surface heat fluxes and the third term on the right-hand side is the adiabatic cooling. C is the common bulk transfer coefficient for sensible and latent heat, |V10| is the 10 m wind, Δr is the difference between radii rc and r0, U is the radial wind and H is the height of the boundary layer. Tess − Te10 is the difference between the equivalent temperature at the sea surface and that at 10 m. Equation (8) is derived in Appendix A. Here we will introduce a non-dimensional parameter,
Substituting Eq. (9) into Eq. (8) gives the following expression for the equivalent temperature of the ascending air:
The denominator is caused by the two-way dependence between the equivalent temperature and the sea-surface heat fluxes.
Equation (10) shows that Λ can be seen as a measure of how efficient the ocean is at heating the cyclone boundary layer. Λ = 0 means that the sea surface does not influence the equivalent temperature of the ascending air at all and the only difference between Tec and Te0 is due to adiabatic cooling. Λ = 1 means that the equivalent temperature of the air entering the cyclone boundary layer has no influence on the equivalent temperature of the ascending air. In this case, Tec will be Tess with the additional effect of adiabatic cooling. Λ < 0 and Λ > 1 are unphysical values since they would indicate that Tec is negatively influenced by Tess and Te0, respectively.
If the boundary-layer pressures, temperatures and humidities are known, then we may solve Eq. (10) for Λ. Once again we take the values from Figure 3 as an example:
If we insert the Λ value from the example above into Eq. (10), we find that Tec ∝ 0.3Tess, i.e. only 30% of the change of Tess will translate into a change of Tec. We conclude that a significant part of the difference between the result from E86b and the result from our numerical simulations can be attributed to boundary-layer physics, which prevents the air from attaining the theoretical maximum effect of an SST change.
4.3. The effect of including the kinetic terms
E86b ignores both the input of angular momentum Wout in Eq. (4) and the change of kinetic energy in the boundary layer Δ(V2)/2 in Eq. (5). The latter is consistent with his assumption that velocities are small outside the cyclone and that convection takes place right at the centre, where there cannot be any horizontal velocity at all. In this subsection we examine the effect of including these terms when estimating the SLP depression sensitivity to SST changes.
E86b shows that if the work to restore the angular momentum takes place at a large radius, it can be estimated as
where f is the Coriolis parameter, r is the radius and V is the azimuthal velocity. We take the values from Figures 2 and 3 as an example. We cannot assume, as E86b does, that the ascent out of the boundary layer takes place right at the centre of the cyclone. We must try to estimate a representative, average, radius of ascent, rc. The ascent from the boundary layer is concentrated in a narrow band with high vertical velocities around 90 km from the centre and a wider band with lower velocities stretching out to 250 km from the centre. Scaled by volume flow, the mean radius of ascent is around 160 km. We assume that the more central band of convection contributes more to the energetics of the cyclone. In an effort to account for this, we put rc = 120 km. We obtain the following estimation for the depression term (DP), the kinetic-energy term (KE) and Wout, all appearing on the right-hand side of Eq. (6):
From the example above it appears unjustified to ignore KE and Wout. If we estimate the internal energy input by balancing Eq. (6), we get quite different results depending on whether these terms are included or not:
The ratio between the two values above is 3.5. The question is how to balance Eq. (3)? If we assume that parameters from the boundary layer are fairly certain, the only remaining uncertainty is the cooling temperature, Tout. If we include KE and Wout we must obviously use a cooling temperature, so that the difference Tb − Tout is 3.5 times smaller than if we exclude these terms. Since E depends linearly on this difference, this means that the SST sensitivity of E will also be 3.5 times smaller. Including KE and Wout while balancing Eq. (3) yields a Tout ≈ −15 °C, where we have used θec = 286 K, consistent with our choice of rc. If we translate this cooling temperature to a height level, we find that it corresponds to about 2 km. That the cooling should take place on a height level well inside the boundary layer is a puzzling result and we will return to this in the discussion section 5.
However, including KE not only alters the calculation of the input from the Carnot cycle, it also produces a closure problem when we alter this input. If KE is ignored in the Bernoulli equation (5), then any change of the input can immediately be translated into a changed SLP depression. If the full Bernoulli equation is used, we must make some assumption about how the changed input will be balanced by changes in DP and/or KE. A possible assumption is that the ratio between the effects on DP and KE should be the same as the original ratio of these terms. Let us test this assumption. We use the SST sensitivities from section 3 and assume a +1 K change of SST:
We see that the ratio between the effects on the two terms is almost identical to the original ratio of the terms, which is at least some vindication for our assumption. If this ratio is to be preserved, any change of DP must actually be larger in magnitude than the change of E by a factor of DP/(DP+KE)=1.4. This moderates the effect on the SST sensitivity of the SLP depression (SPSST) of including KE and Wout in the estimation of E.
The combined effect of including KE and Wout and preserving the ratio of the Bernoulli terms is that the SPSST must be multiplied by a factor of 0.4.
5. Summary and discussion
We run the numerical model with identical atmospheric initialization and different SSTs. It is, unsurprisingly, found that higher SSTs tend to generate more intense cyclones, with deeper SLP depressions and stronger azimuthal winds. When we perturb the SST, the cyclones respond with intensification (decay) for positive (negative) perturbations. On average the response is proportional to the size of the perturbation. The SLP depression deepens by about 0.6 hPa K−1 and the maximum azimuthal wind increases by about 0.6 m s−1 K−1.
Saetra et al. (2008) show that strong wind events over the Nordic Seas, such as polar lows, can lead to an SST increase by up to 2 °C within some hours. If we assume that the polar lows are similar to the ones simulated in this study, we would expect them to deepen by 1 hPa. That is, however, not large compared with the observed variations a polar low can undergo during even shorter time-scales.
Several studies, e.g. Bender et al. (1993), Shade and Emanuel (1999), Sutyrin and Khain (1984), have shown that the decreasing SST associated with tropical cyclones has a limiting effect on their intensity. It would be an interesting continuation of the present study to take the analytical approach suggested here to estimate the SST sensitivity of cyclone intensity and apply it to a tropical environment.
The effect of the SST perturbations on the intensity of the simulated polar lows is considerably weaker than we anticipated. The effect is about one order of magnitude less than E86b estimates for the effect on the theoretical maximum intensity. Emanuel uses the WISHE theory, in which polar lows are assumed to obtain a substantial part of their energy from a Carnot cycle. Building upon the work of Emanuel, we derive an analytical method for predicting how sensitive a polar low is to changes in the SST. The method is constituted from two main parts:
(1)one part for estimating how efficiently the polar low extracts heat from the ocean and
(2)one part for estimating the temperature during the cooling phase of the Carnot cycle.
Our starting point is a known state near the sea surface in a polar low, exemplified by one of the control runs of our numerical simulations. In the first part we derive a non-dimensional efficiency parameter, Λ. Physical values for Λ range from zero to one. Λ = 0 means that the sea-surface heat fluxes have no influence on the air. Λ = 1 means that any change of the SST (and the sea-surface humidity) is transferred to the air with the same magnitude. In our example only 30% of an SST change is transferred to the air.
In the second part we estimate the cooling temperature by balancing the energy from the Carnot cycle against the work in the polar low. This work is both the friction in the polar-low surface layer and the work done by the ambient atmosphere to supply the polar low with angular momentum, which is lost in the surface layer. The latter work is thus contributing to overcome the friction, just as the energy from the Carnot cycle is. The boundary-layer friction is balanced by the depression and the kinetic-energy change, as stated in the Bernoulli equation. In his calculations E86b ignores the work done by the ambient atmosphere, as well as the kinetic-energy change. At least in our example, this appears unjustified, as we estimate both the input from the ambient atmosphere and the kinetic-energy change to be the same order of magnitude as the input from the Carnot cycle.
E86b estimates the cooling temperature by following the moist adiabats in the polar low from near the surface up and outwards to a large radius, yielding a value close to the tropopause temperature. Our method yields a considerably higher temperature, −15°C, as opposed to −58°C at the tropopause. This implies a lower contribution from the Carnot cycle to the polar-low intensity, which results in a lower SST sensitivity. It also implies that the mean height of convection is considerably lower. In our example we estimated the height corresponding to the mean cooling temperature to be about 2 km. This is well inside the boundary layer, which we find puzzling. However, we should remember that the cooling may take place during a slow descent corresponding to a considerable range in both height and temperature. The top of the (mean) convection may very well reach 4 km. We should also remember that our method for calculating Tout captures the mean cooling temperature of an average parcel. Even though it is apparent that some convection reaches above 7 km, it is likewise apparent that much of the convection does not. We propose that a considerable part of the boundary-layer air moves in a rather small Carnot cycle. Our method for estimating Tout can be seen as a way of estimating the vertical (or temperature) extent of the average Carnot cycle.
The inclusion of the kinetic-energy term in the Bernoulli equation introduces a closure problem, which we propose can be solved by assuming a constant relationship between the kinetic-energy term and the depression term. This moderates the first effect of including the input from the ambient atmosphere and the kinetic-energy change, which was a warmer Tout. The total result of our method is a good prediction of the order-of-magnitude difference between the sensitivity from our simulations and the sensitivity of the theoretical maximum intensity estimated by E86b.
From the present investigation, it appears that the significant variations in observed intensity of real polar lows are difficult to explain by variations in the SST, at least within the idealized axisymmetric WISHE framework. One possibly important parameter is the static stability of the atmosphere. An often-used indicator of the static stability is the difference between the SST and the temperature at the 500 hPa level (T500). SST C is sometimes considered a condition favourable for polar-low development (Zahn and von Storch, 2008; Blechschmidt et al., 2009). However, it not clear whether lower static stability (larger value of SST −T500) leads to higher intensity or whether its importance is only as a necessary precondition for the development of polar lows. The latter is more in line with WISHE, where a cornerstone is the assumption of a nearly neutral vertical profile. Emanuel and Rotunno (1989) tried varying the initial vertical temperature/humidity profile and found that the main impact was on the timing of the onset of intensification once the surface fluxes have established a near-neutral convecting atmosphere.
There are of course limitations to what can be studied in the approach adopted herein, and we close this section with a brief discussion about the generality of our results. Some processes can only be described in a full three-dimensional model. Tropical cyclones generally show a higher degree of (large-scale) axisymmetry than polar lows. Horizontal translation nevertheless leads to stronger wind and higher sea-surface fluxes in the front right-hand side quadrant of the cyclone (in the northern hemisphere and looking in the direction of translation). Kafatos et al. (2006) found that high SST anomalies in this quadrant were of particular significance for the intensification of HurricaneKatrina (2005). Many polar lows have strong asymmetries in structure and it is possible that they are significantly more sensitive to SST changes, e.g. in conjunction with a spatial variation of the SST. In the observed polar low reported by Saetra et al. (2008), the largest change in the SST take place to the right of the polar-low track centreline. It is thus possible to imagine an atmosphere–ocean interaction that would require a coupled model to capture. However, we believe that our approach with an idealized model and the inclusion of one-way action only is useful, especially as a starting point for an estimation of the relative importance of different parameters and processes.
We are aware of two investigations in which a three-dimensional numerical model has been used to simulate polar lows at different SSTs. Blier (1996) use a mesoscale model (PSU-NCAR MM4) to simulate an observed ‘polar airstream’ cyclogenesis over the Gulf of Alaska. This cyclonic development was strongly asymmetric and, at least initially, very influenced by a cold upper-level disturbance moving over a low-level baroclinic zone. Changing the SST of his control run by +4 °C, he obtains a central SLP that is deeper by 8 hPa.
More pointedly, the very small SST sensitivity of our simulations contrasts to some extent with the results of Albright et al. (1995). They also use the PSU-NCAR MM4, simulating an observed polar low over Hudson Bay. Changing the SST of their control run by +8 °C, they obtain a central SLP that is deeper by 16 hPa. They stress that baroclinic processes contributed only a little to the development. However, the Hudson Bay polar low occurred over a small patch of open water surrounded by sea ice and land. In this case, the gradient in surface heating across the boundary between open ocean and ice/coastline will be greater than if open ocean were imposed over the whole domain. Therefore one might expect a larger sensitivity to SST. This situation is of course not relevant to our area of interest over the North Atlantic Current.
The contribution from Torsten Linders was funded by the European Commission through a research fellowship under the ModObs project, contract MRTN-CT-2005-019369, and by the Research Council of Norway through the project IPY-THORPEX, contract 175992/S30. Øyvind Saetra was funded by the Research Council of Norway through the projects ArcChange, contract 178577/S30, and IPY-THORPEX, contract 175992/S30. We thank Professor Jan Erik Weber for support and valuable discussions.
A. Derivation of the expressions for E and Tec
The rate of entropy change, J/T, of an air parcel is given by Holton (2004, sections 2.7.1 and 9.5.1):
where J is the rate of energy change and D/dt is the material derivative. The energy received in the boundary layer is found by integrating Eq. (A1) from the point where the air enters the cyclone boundary layer to the point where it ascends out of the boundary layer (from radius r0 to radius rc in Figure 6):
where Tb is the mean temperature during the heating. Performing the same integration over the remaining parts of the Carnot circuit (from radius rc to radius r0 in Figure 6) gives the energy loss during cooling:
where Tout is the mean temperature during the cooling. Adding Eqs (A2) and (A3) gives the total net internal energy input, expressed in Eq. (3).
The temperature change, DT, of an air parcel is given by Holton (2004, section 2.7):
where Dp is the pressure change and dt is the time differential. In the boundary layer the rate of energy change is essentially the heat flux from the sea surface. If both the sensible and latent heat fluxes are included, then Eq. (A4) is an expression for the rate of change of the equivalent temperature, Te, which we define as
where Lv is the latent heat of vaporization and q is the specific humidity.
To find the rate of energy change, we have to divide the total surface heat flux, Q, by the air density and the boundary-layer depth H over which the heat flux is distributed:
where we have exchanged the time differential, dt, for the radial differential, dr, by using
where U is the radial velocity. We integrate Eq. (A8) radially across the boundary layer (from radius r0 to radius rc in Figure 6). We obtain Eq. (8), where we now let the parameters on the right-hand side represent mean values.
The alternative would be to have separate expressions, similar to Eq. (10), for sensible and latent heat as well as introducing an additional parameter for condensation. For the values in Figure 3 our method underestimates the equivalent potential temperature, but never by more than 0.14%.