1. Introduction
It is always a goal within the climate and weather prediction communities to seek more realistic representations for atmospheric processes. Improvements are sought for the processes typically referred to as physical “parameterizations” (those processes dominated by space and time scales far below the resolution of large-scale atmospheric models), and on the numerical techniques used to solve the differential equations for the fluid dynamics of the atmosphere (frequently referred to as the “dynamical core”). Occasionally, modeling efforts have also focused on new approaches in formulating model equations for fluid flow in different coordinate systems and on the investigation of their impact on the climate simulations. The advancement in dynamical cores has been proven to play an important role in the improvement of GCM climate simulations.
While most model errors may be random, some model biases appear to be systematic and universal among most GCMs. One example is the general cold bias problem found in the climates simulated by most GCMs. In an intercomparison of 14 GCMs, Boer (1992) noted that it was remarkable that all models tended to simulate temperatures colder than observations even though different numerical methods, physical parameterizations, and resolutions were used. This cold bias was found to be most pronounced in the upper troposphere poleward of 50° in both hemispheres where a negative temperature bias of more than 10°C could be found. The same model bias is present in today’s climate models as seen in the simulations discussed in Solomon et al. (2007). Johnson (1997) presented a theoretical study that provided an explanation for this shared deficiency in most GCMs. He argued that the bias originated from the choice of model coordinates that permitted the generation of nonphysical sources of entropy and that a model coordinate able to eliminate aphysical sources of entropy would be required to remove this cold bias problem in any GCM.
In recent years, extensive modeling efforts by Lin and colleagues (see Lin 2004) have led to the development of a numerical solution for the equations of motion that we refer to hereafter as a finite-volume (FV) dynamical core. The FV core employs several innovative numerical techniques. The FV core employs a semi-Lagrangian flux-form transport scheme (Lin and Rood 1997) for the horizontal transport terms, to provide a mass-conserving solution stable for high advective Courant number flows, formulated under a standard latitude–longitude grid. Additionally, the piecewise parabolic method (PPM) (Colella and Woodward 1984; Carpenter et al. 1990) is incorporated in the transport scheme to obtain highly accurate, shape-preserving solutions in the presence of sharp gradients and smooth flows (Carpenter et al. 1990). Nevertheless, it is worth noting, though, that the monotonicity constraint of the transport scheme in the FV core is formulated in one dimension but does not guarantee monotonicity in two-dimensional flows.
Following the concept proposed by Starr (1945), the model equations of the FV dynamical core were formulated using a Lagrangian vertical coordinate (Lin 2004). Making use of a Lagrangian vertical coordinate allows the FV core to reduce fully 3D atmospheric transport to semihorizontal (2D) equations. During a numerical simulation, model layers may deform dramatically, and a procedure, called “vertical remapping,” is performed periodically to rearrange model variables to a prescribed coordinate system. This step implicitly accounts for vertical transport within atmospheric dynamics. We will discuss this in more detail below. Hence, the FV formulation with a Lagrangian coordinate reduces the coupling among vertical layers significantly and consequently the numerical accuracy of the representation is greatly improved.
The definition of the vertical coordinate under the concept of Lagrangian vertical coordinates can play a crucial role in the numerical accuracy of the core. Under adiabatic conditions the fluid moves along isentropes and no transport across isentropes is experienced. The FV core exploits the concept of separating the diabatic forcing terms from the adiabatic flow, first producing a solution ignoring the diabatic terms, and then accounting for them afterward. By ignoring the diabatic terms the FV core only handles adiabatic motions. If the model layers were defined by a dynamically conserved variable, that is, corresponding to isentropes, the time integration of the thermodynamic equation becomes trivial and higher accuracy is thus obtained.
The importance of vertical discretization in atmospheric models has a long history (Kasahara 1974; Sundqvist 1976) for climate and weather prediction problems. Currently the most widely used vertical coordinate in GCMs is pressure based (σ or σ − p coordinates). These coordinate systems produce equations that are inherently 3D in nature. More recently, a new perspective in formulating model equations, that is, isentropic modeling, has been proposed (Benjamin 1989; Hsu and Arakawa 1990). In an adiabatic atmosphere the airstream flows along isentropes. Under a model formulated in an isentropic coordinate, without any diabatic processes, one should observe two-dimensional flows since the vertical component of atmospheric motions vanishes. In the upper troposphere and above, the atmospheric flows are mostly 2D due to the rarity of intense diabatic processes. Therefore, the isentropic modeling framework provides potential advantages in accurately simulating general circulations of the atmosphere. Nevertheless, there are some disadvantages when a purely isentropic coordinate is utilized and they have been discussed in Webster et al. (1999) and Arakawa (2000).
One common problem found in a purely isentropic coordinate model is the coarse vertical resolution in the equatorial troposphere. Since the vertical resolution in an isentropic coordinate is determined by the stratification of the atmosphere, an isentropic model typically suffers from a very coarse vertical resolution in the equatorial troposphere where the atmosphere tends to be least stable. Another commonly found issue in a purely isentropic coordinate system is that the lowest isentropes tend to intercept the terrain. One common solution to this issue is to allow massless layers near the earth’s surface.
To remedy the disadvantages found in a purely isentropic coordinate, several studies have proposed a hybrid-isentropic coordinate (σ − θ coordinate) approach and its effectiveness has been demonstrated in several applications. In an offline calculation, Mahowald et al. (2002) demonstrated that a hybrid-isentropic coordinate was able to offer a more accurate solution than a pressure-based vertical coordinate in terms of tracer transport, especially in regions where severe diabatic heating associated with convection rarely took place. In addition, Zhu and Schneider (1997) and Webster et al. (1999) showed great improvements for the simulated atmospheric circulations in the upper troposphere and stratosphere by using a σ − θ coordinate. Furthermore, Schaack et al. (2004) presented a 14-yr climate simulation with a hybrid isentropic model in which the general cold bias problem was significantly reduced. Their results imply that the characteristics of heat transport of a GCM can be quite sensitive to the choice of vertical discretization and are consistent with the theoretical study by Johnson (1997).
In this paper, we describe the implementation of a hybrid isentropic coordinate in the FV dynamical core, which is now the default option in the Community Atmosphere Model (CAM) at the National Center for Atmospheric Research (NCAR) along with Eulerian spectral (EUL) and semi-Lagrangian dynamical cores. In this paper we compare a 20-yr climate simulation with the isentropic finite-volume (IFV) dynamical core coupled with the NCAR CCM3 physics suite with the simulation by the FV model using the traditional σ − p coordinate. Our modeling work presents the first study that incorporates the ideas of a vertically Lagrangian coordinate and an isentropic model formulation.
2. The NCAR isentropic finite volume model
Table for a 26-layer hybrid-isentropic coordinate vs a 26-layer hybrid-isobaric coordinate of CAM (27 interfaces). The last column shows typical pressure for the hybrid-isobaric coordinate by assuming a surface pressure of 1000 hPa.
A pressure-based vertical coordinate is not an ideal choice for accurate simulations of tracer transport processes. In the atmosphere there is always forcing that excites internal gravity waves, which continuously perturb the pressure field. The pressure field is perturbed, but atmospheric pressure is not a dynamically conserved variable. As a result, some undesirable vertical transport can be introduced during the vertical remapping procedure. However, if a variable is chosen as the vertical coordinate that approximates a material surface, such nonphysical vertical transport can be avoided. In the atmosphere potential temperature serves this purpose well since it is a conserved variable under adiabatic conditions. Additionally, it is typically monotonic in the vertical; that is, the atmosphere is statically stable.
Our formulation for the vertical coordinate is similar in many regards to that discussed in Schaack et al. (2004), with some notable differences discussed below. The atmosphere is divided into three zones. For the atmosphere above θ = 336 K, the model interfaces follow a set of prescribed isentropes. One layer, around 10 hPa deep, is inserted at the surface to ensure adequate representation of the planetary boundary layer. A transitional zone is constructed between the surface layer and the isentropic zone.
The values of ξ used in the NCAR isentropic finite volume (IFV) dynamical core are listed in Table 1. There are 12 isentropic layers (13 model interfaces), 13 transitional layers, and 1 purely hybrid surface layer. The model top is set to be isobaric, that is, p(ξt) = 2.194 067 hPa, as done in the FV model. This ensures that no source or sink attributed to the upper boundary condition for the total mass of the entire computational domain is present.
The difference between the hybrid pressure-based coordinate and the hybrid isentropic coordinate in the NCAR CAM is quite evident as illustrated in Fig. 1. The most dramatic difference between them lies in the upper troposphere and stratosphere. In this region, the pressure-based coordinate becomes mostly isobaric and, thus, each layer possesses a uniform mass field. On the other hand, the hybrid isentropic coordinate gradually transits to be purely isentropic in this region and the model interfaces are solely determined by the positions of various isentropes. As a result, the mass within a layer varies between pole and equator. Because of the high stability above the tropopause, layers tend to be very tightly spaced there, especially at high latitudes. The increase in resolution near the tropopause is balanced by a coarser resolution in the upper stratosphere. In the lowest atmosphere both coordinate systems are terrain following in nature and thus a smaller difference can be detected between the two coordinate systems.
Model interfaces for (a) the IFV core (σ − θ), (b) the IFV380 core (σ − θ), (c) the IFV500 core (σ − θ), and (d) σ − p coordinates at 97.5°E at the initial time. The thick black line in (a),(b), and (c) represents the 336-, 380-, and 500-K isentrope, respectively; the interface between the transitional domain and isentropic domain. The dashed lines in (a),(b), and (c) denote various isentropes selected for the hybrid isentropic coordinate in the isentropic domain.
Citation: Journal of Climate 25, 8; 10.1175/2011JCLI4184.1
The large variations in mass within a layer can produce computational difficulties in an isentropic model. Schaack et al. (2004) reported that some negative values could be introduced in the mass field within the isentropic domain during their model simulation. To remedy this problem, they employed a global mass borrowing technique to guarantee the mass of each grid cell to be positive. With the transport scheme currently implemented in the FV model, we have experienced the same negative mass issue. As described in Lin and Rood (1996), the transport scheme of the FV core is strictly monotonic only in one dimension, but overshoots and undershoots are possible for two-dimensional transport. We have implemented a positive-definite limiter, following Skamarock (2006), that adjusts the fluxes across the boundaries of a grid cell to ensure the integrated scalar field remains positive. The detail with regard to the positive definite limiter is provided in the appendix.
The University of Wisconsin θ − η model (Schaack et al. 2004) did not assume adiabatic flow, and explicit vertical transport terms appeared in the equations associated with diabatic processes. This information drives the model interfaces up/down when there is cooling/heating occurring, which in turn induces vertical transport. Because our IFV core employs a vertically Lagrangian coordinate and neglects the heating terms during that component of the model time step, an alternate strategy is needed.
Our numerical integration is advanced using the following substeps.
We assume the variables are defined between “reference surfaces.” Equations (1)–(5) are used to update the state variables (u, υ, δp, θv, q) with the quasi-Lagrangian tendencies. These fields no longer lie within the coordinate surfaces defined above.
The state variables are remapped in the vertical back to the reference surfaces.
The physical parameterizations are called to update the state variables. The fields again no longer lie on the reference coordinate surfaces.
A vertical remapping is again employed.
The information is written out to a file when appropriate, and these steps are repeated.
Strictly speaking, the only difference between the FV and IFV model is the choice of the reference surfaces to which all model are remapped. In CAM simulations, vertical remapping takes place every 30 min. For the IFV model, the first vertical remapping, substep 2, is not necessary. For simplicity, it is kept to hold the overall model structure since it is where vertical remapping takes place in the FV model. Nevertheless, the remapping in substep 4 is required in the IFV model, which is an addition to the FV model.
The vertical remapping process accounts for vertical transport. The remapping is performed by first producing a subgrid distribution of potential temperature using a piecewise parabolic method (PPM) from the layer mean temperature fields within each model layer. Then the model searches for the position of the identified isentropes (or value of ξ) where it places the new model interfaces. The mean value of the field within each new set of layers is then computed from the subgrid-scale distributions. The search and remap procedure is straightforward since, within each model layer, the PPM algorithm creates a subgrid distribution that is quadratic in pressure and monotonic. Therefore, a solution for the altitude of a specific isentrope can be uniquely obtained by a simple quadratic formula. It is worth noting that the accuracy of the numerical algorithm used to construct a subgrid distribution becomes crucial for this task. As a result, some vertical transport could be expected in the IFV model even when there is no diabatic processes taking place since a perfect solution in positioning the isentropes is needed to avoid any spurious vertical transport that most algorithms will fail to achieve. We have found that minimal vertical transport is introduced by this procedure within the isentropic zone when the flow is purely adiabatic.
3. An IFV climate simulation
a. Zonally averaged temperature distribution
We first focus our attention on the temperature field simulated by the IFV model. As illustrated in Fig. 2, the IFV model produces a realistic zonal average distribution of annual average (ANN) temperature as well as seasonal means in December–February (DJF) and June–August (JJA).
Zonally averaged atmospheric temperatures simulated by the IFV model: (a) the DJF mean, (b) the JJA mean, and (c) the ANN mean.
Citation: Journal of Climate 25, 8; 10.1175/2011JCLI4184.1
To evaluate the performance of the IFV model in simulating the atmospheric temperature field, the temperature difference field or temperature bias was computed between that simulated by the three dynamical cores examined in this paper and the European Centre for Medium-Range Weather Forecasts (ECMWF) reanalysis (Kållberg et al. 2004). As illustrated in Fig. 3a, the spectral core (EUL) produces a dramatic temperature bias more than 10°C colder in the DJF mean. The most pronounced cold bias is found poleward of 50°S between 100 and 300 hPa. On the other hand, substantial cold biases of up to 7°C are also present poleward of 60°N. In addition, a weak cold temperature bias can be found in the upper troposphere and lower stratosphere in the midlatitudes and subtropics/tropics.
Zonally averaged mean (DJF, JJA, and ANN) atmospheric temperature bias against the ECMWF reanalysis simulated by the EUL, FV, and IFV models.
Citation: Journal of Climate 25, 8; 10.1175/2011JCLI4184.1
As shown in Fig. 3d, the FV model creates a very similar temperature bias distribution as that produced by the EUL model. Some features in common between the two models include the pronounced polar cold bias and the slightly colder upper troposphere and lower stratosphere in the subtropics and equatorial regions. Furthermore, both models generally show stronger cold bias around the summer pole. Nevertheless, it is evident that the FV model does a better job in simulating atmospheric temperatures than the EUL model. The largest improvements were found near both poles where the area occupied by cold temperature bias became slightly smaller.
The IFV model exhibits further improvements in the simulated temperature field over the FV model. First, the cold bias found in the summer polar region continue to shrink in extent (Fig. 3g). Furthermore, the region where cold bias is previously present around the winter pole in the EUL and FV models now show very small temperature bias, of less than 3°C, in the IFV model, and it no longer extends as low in altitude. Consequently, the temperature field simulated by the IFV model in the low and midtroposphere near the winter pole is slightly colder than in the EUL and FV models. In the subtropics and equatorial region the cold bias found earlier in the EUL and FV cores is almost totally eliminated in the IFV model. Instead, the atmospheric temperature is even slightly warmer than the ECMWF reanalysis. However, an undesirable feature is introduced by the IFV model: cold bias up to 5°C in the equatorial region around 70 hPa is introduced by the IFV model. Nevertheless, as evidenced by the very different atmospheric temperature distribution found in the IFV model, the IFV model may simulate a significantly different general circulation of the atmosphere. This is confirmed by the improvement in the sea level pressure distribution over the North Hemisphere by the IFV model, which will be discussed later.
In JJA (shown in Figs. 3b,e) the EUL and FV cores produce a similar temperature bias pattern as found in DJF; that is, a very pronounced cold bias is again found around the summer pole. Additionally, the EUL core also simulates intense cold bias in the winter pole. The magnitude of the cold bias at both poles is almost equally pronounced, exceeding 10°C, and the cold bias extends over a large area around both polar regions (Fig. 3b). The FV core again shows some improvement over the EUL core in reducing the intensity of cold temperature bias in the polar regions down to ~7°C (Fig. 3e). On the other hand, both models produce a similar temperature bias pattern, up to 4°C colder, in the subtropics and equatorial region centered at around 100 hPa.
The IFV model shows its biggest improvement in the winter pole where the maximum temperature bias is reduced to 4°C colder (Figs. 3e,h) and the area occupied by cold bias is also highly reduced. The IFV model shows less improvement in the summer polar region and even slight degradation can be found, that is, near 60°S at 250 hPa where the temperature becomes colder than that simulated by the FV model. Consistent with the EUL and FV cores, the IFV model also produces a colder troposphere and lower stratosphere in the subtropics and equatorial region, but the intensity of the maximum cold bias is enhanced. This undesirable attribute of the IFV model requires further investigation.
For the ANN mean temperature field, the contributions from the seasonal means of DJF and JJA are to be expected with some additional contributions from the spring and fall months. Hence, it is anticipated that the ANN mean cold temperature bias in the polar regions is not as intense (Figs. 3c,f,i) as the seasonal means of DJF and JJA since the ANN mean tends to moderate the more intense cold bias found in the winter pole. As found in the seasonal means in DJF and JJA, the FV core shows some improvement in decreasing the cold bias in the polar regions over the EUL core (cf. Figs. 3c and 3f) for the ANN mean. Less improvement is found in the subtropics and equatorial region where the temperature bias of the EUL and FV cores in the upper troposphere and lower stratosphere exhibits a very similar pattern and both remain almost equally colder than the ECMWF reanalysis. The IFV model shows further improvements over the FV model by reducing both the magnitude of temperature bias around both poles and the area occupied by cold bias. Furthermore, the lower portion of the region where small cold bias is previously present in the subtropics and equatorial region in the EUL and FV cores is no longer too cold. Some small warm bias is even found in this region. Nevertheless, more pronounced cold bias occurs at higher altitudes centered at 70 hPa in the IFV model, as found previously in the seasonal means of DJF and JJA.
To contrast the simulated atmospheric temperatures between the FV and IFV models, the temperature difference field between the two models was calculated and illustrated in Fig. 4. Diagnosed by the seasonal means of DJF (Fig. 4a) and JJA (Fig. 4b), it is clear that overall the IFV simulates a warmer upper troposphere and lower stratosphere throughout the globe than the FV core. The most pronounced difference is found near the winter polar region, which extends from the upper troposphere into the stratosphere (reaching the top of the computational domain). In the polar upper stratosphere the air temperature is up to 10°C warmer than that simulated by the FV model. At the summer pole the warmer air temperature is less intense and is only confined in a thin layer centered at around 200 hPa. Additionally, the IFV model also simulates slightly warmer temperatures in the upper troposphere and lower stratosphere in midlatitudes and the subtropics. Still, the ANN mean (Fig. 4c) follows the overall pattern as the seasonal means of DJF and JJA, but the signals are less pronounced. The warmer air temperature simulated by the IFV core enables the model to generate atmospheric temperatures much closer to observations. This has great influence on other aspects of the simulation, which will be discussed later.
Zonally averaged atmospheric temperature difference between the IFV and FV models for: (a) the DJF mean, (b) the JJA mean, and (c) the ANN mean.
Citation: Journal of Climate 25, 8; 10.1175/2011JCLI4184.1
b. Zonally averaged specific humidity, relative humidity, and cloud distribution
By changing the vertical discretization of a dynamical core, the characteristics of tracer transport induced by atmospheric dynamics can be substantially modified. This is manifest by the different distribution of water vapor simulated by the FV and IFV models. It is worth noting that the two models utilize the same physical parameterizations and that the major difference between them is merely the vertical discretizations. Therefore, the difference in the fields simulated by the two models may be mainly attributed to the vertical coordinates. Figure 5a shows the ANN mean specific humidity simulated by the FV core, and it is interesting to note that the biggest difference between the IFV and FV models is in the low and midtroposphere (Fig. 6a) where the two model coordinates are very similar (terrain following). These results reveal that the IFV model tends to preserve more water vapor in the lowest troposphere than the FV model, mainly in the equatorial region. However, the IFV model makes the midtroposphere (around 700 hPa) drier. Minimal difference is found above 600 hPa between the IFV and FV models.
Zonally averaged fields of annual mean simulated by the FV model: (a) specific humidity in g kg−1, (b) relative humidity, and (c) cloud fraction.
Citation: Journal of Climate 25, 8; 10.1175/2011JCLI4184.1
Difference in zonally averaged annual-mean specific humidity (top) in grams per kilogram and (bottom) in percentage from the FV model simulated by (a) the IFV model, (b) the IFV380 model, (c) the IFV500 model, (d) the IFV model, (e) the IFV380 model, and (f) the IFV500 model.
Citation: Journal of Climate 25, 8; 10.1175/2011JCLI4184.1
Since the temperature and water vapor fields simulated by the IFV model are very different from those in the FV model, it is likely that the relative humidity field can also be significantly modified. As illustrated in Fig. 7a, the ANN mean relative humidity difference field shows that the IFV model tends to produce a drier upper atmosphere centered at 200 hPa throughout the globe. While the specific humidity field simulated by the IFV core shows little difference in magnitude at this altitude (Fig. 6a), the difference in percentage is substantial (Fig. 6d): a reduction of up to 60% is found in midlatitudes and an increase of up to 40% is present in low latitudes. Therefore, the lower relative humidity in low latitudes cannot be explained by the distribution of specific humidity, but can only be induced by the temperature difference. This may be confirmed by comparing Fig. 7a with Fig. 4c: the regions with dryness in the upper troposphere in Fig. 7a show warmer atmospheric temperatures in Fig. 4c and these two fields show outstanding agreement. Similarly, since the IFV model produces an undesirable cold bias in the equatorial region at high altitude centered at 50 hPa (Fig. 4c), this serves to enhance the relative humidity in this region. Since in the upper troposphere at ~400 hPa the IFV model does not produce much warmer air temperature compared with the FV model, the increase in relative humidity in these regions must be attributed to slightly more water vapor (see Fig. 6a). This feature may be viewed as a signal that it is more difficult for tracers to move into the isentropic zone from the transitional zone. This can only be achieved if the tracers penetrate the 336-K isentrope in the IFV model. Therefore, cross-isentropic water vapor transport is reduced in the transitional zone in the IFV model. In the lowest troposphere, especially in the equatorial region, the atmosphere possesses a higher relative humidity. This feature is consistent with the specific humidity field.
As in Fig. 6, but for relative humidity.
Citation: Journal of Climate 25, 8; 10.1175/2011JCLI4184.1
Here we present quantitative calculations to determine if the change in temperature is indeed the key factor driving the difference in relative humidity as stated earlier. In Fig. 8a, we assume that the ANN temperature from the FV core is coupled with the ANN specific humidity from the IFV core. This scenario examines the impact of specific humidity on the relative humidity field. In Fig. 8b, we assume that the ANN temperature from the IFV core is coupled with the ANN specific humidity from the FV core. This scenario examines the impact of temperature on the relative humidity field. The results clearly reveal that temperature plays a more important role in reducing relative humidity across the globe at 200 hPa. Nevertheless, the IFV core introduces some nonnegligible moistening in low latitudes near 150 hPa (see Fig. 8a). One detail to note is that the offline calculation of relative humidity by using ANN temperature and specific humidity will not agree with the relative humidity from the model output calculated from instantaneous values at each time step.
Relative humidity difference from the FV model by using (a) specific humidity from the IFV core and temperature from the FV core, (b) specific humidity from the FV core and temperature from the IFV core, (c) specific humidity from the IFV380 core and temperature from the FV core, (d) specific humidity from the FV core and temperature from the IFV380 core, (e) specific humidity from the IFV500 core and temperature from the FV core, and (f) specific humidity from the FV core and temperature from the IFV500 core.
Citation: Journal of Climate 25, 8; 10.1175/2011JCLI4184.1
Cloud cover plays an important role in simulating the general circulation since it affects the local radiative balance by altering the amount of incoming shortwave radiation and outgoing longwave radiation. Therefore, the cloud distribution has a prominent impact on climatology. In most GCMs, the relative humidity of air is an important parameter in determining cloud fraction. Thus, as described earlier, the simulated cloud distribution can be substantially different in the IFV model since the relative humidity field is significantly modified. Indeed, the most noticeable difference in the cloud distribution field simulated by the IFV model is a large reduction in the amount of high clouds (Fig. 9a), most pronounced in the equatorial region centered at 200 hPa. This region is collocated with a large reduction in relative humidity mainly due to the warmer air temperature introduced by the IFV model. Directly above and below this region where a reduction in cloud fraction is found, an increase in cloud fraction is present. This pattern is also in good agreement with what the distribution of relative humidity implies. The reduction of the amount of high clouds by the IFV model is a very desirable feature since the amount of high cloud simulated by the FV and spectral cores is well above observations (Lin and Zhang 2004). In the equatorial region below 850 hPa an increase in cloud fraction is also present. As found previously, more water vapor is found around this region in the IFV model and this leads to a higher relative humidity that, in turn, enhances cloud fraction in this region.
As in Fig. 6, but for cloud fraction.
Citation: Journal of Climate 25, 8; 10.1175/2011JCLI4184.1
c. Mean sea level pressure
One deficiency shared by the EUL and FV cores of NCAR CAM is their poor performance in simulating the DJF mean sea level pressure in the Northern Hemisphere. Both tend to produce large bias in the DJF mean sea level pressure field. The National Centers for Environmental Prediction (NCEP) reanalysis (Kistler et al. 2001), as illustrated in Fig. 10d, displays some key features in the DJF mean sea level pressure field of the Northern Hemisphere, including the Icelandic low over the North Atlantic and the Aleutian low over the eastern Pacific. As shown in Figs. 10a,b, the Icelandic low simulated by both EUL and FV cores is much deeper than the NCEP reanalysis, although the EUL core does a slightly better job. On the other hand, the Aleutian low is significantly underestimated by both models. Additionally, both cores also displace the Aleutian low poleward and westward from the NCEP reanalysis. These large mean sea level pressure biases typically lead to an inaccurate surface wind field, which is responsible for driving upper-ocean circulations.
DJF mean sea level pressure (hPa) in the Northern Hemisphere simulated by (a) the EUL model, (b) the FV model, (c) the IFV model, and (d) by the NCEP reanalysis.
Citation: Journal of Climate 25, 8; 10.1175/2011JCLI4184.1
The DJF mean sea level pressure field simulated by the IFV model (Fig. 10c) shows great improvements over both EUL and FV models. The most significant improvement the IFV model makes is the reduction in strength of the Icelandic low compared with that simulated by the EUL and FV models, even though the IFV model still slightly overestimates its intensity. Another big improvement that the IFV model makes is its ability to better simulate the intensity of the Aleutian low. As seen in Fig. 10c, the IFV model does a much better job than the FV model by raising the strength of the Icelandic low and bringing its intensity much closer to the NCEP reanalysis. The IFV model also exhibits a better performance in simulating the intensity of the Aleutian low when compared with the EUL model even though the improvement is smaller. However, the IFV model clearly displaces the Aleutian low even closer to the pole than the EUL model. One feature common in the three cores is that they all tend to place the Aleutian low too far westward compared to the NCEP reanalysis.
4. Alternate modeling approaches to reduce cold bias
In this section, we explore other model configurations as an attempt to reduce the general cold pole problem in GCM simulations. The first approach is to utilize a pressure-based vertical coordinate but to increase the resolution between 150 and 300 hPa with the FV core. The second approach is to use the hybrid-isentropic coordinate system but to lift the isentropic zone above where the cold pole problem is present.
a. A 35-layer hybrid-isobaric model
Since the IFV model drastically increases the vertical resolution in the lower stratosphere where the cold bias is found near the poles, one may speculate that the reduction of cold bias is simply a by-product of an enhanced vertical resolution in this region. To further investigate this matter, we insert nine extra layers between 150 and 300 hPa in the default CAM hybrid-isobaric coordinate, which makes the average depth of each layer in this zone to be ~10 hPa. The detail of this model configuration (hereafter FVL35) is found in Table 2.
Table for a 35-layer hybrid-isobaric coordinate (36 interfaces). The last column shows typical pressure for the hybrid-isobaric assuming a surface of 1000 hPa.
The temperature bias simulated by FVL35, illustrated in Fig. 11 shows minimal difference from the FV model (see Figs. 3d–f for comparison). A cold bias > 10°C has returned even though the vertical resolution between 150 and 300 hPa is highly enhanced. Hence, increasing vertical resolution in the pressure-based coordinate system merely increases the computational cost but achieves very little in reducing the cold bias.
Zonally averaged mean (DJF, JJA, ANN) atmospheric temperature bias against the ECMWF reanalysis simulated by the FVL35, IFV380, and IFV500 models.
Citation: Journal of Climate 25, 8; 10.1175/2011JCLI4184.1
b. Two hybrid-isentropic model with different isentropic zones
The previous experiment implies that enhancing the vertical resolution of a pressure-based coordinate system is not capable of reducing the cold bias. We now return to the isentropic modeling approach, but raise the isentropic zone to higher altitude. Two experiments are conducted and the isentrope defining the isentropic zone is lifted to 380 (hereafter IFV380) and 500 K (hereafter IFV500). The detailed configuration of these two isentropic models can be found in Table 3.
Table for two hybrid-isentropic coordinate systems: IFV380 and IFV 500.
Overall, the IFV380 performs similarly to the IFV model. However, the IFV380 loses some improvements seen in the IFV model. As illustrated in Fig. 11d, the cold bias in DJF near the North Pole in IFV380 maintains a similar magnitude as in IFV (Fig. 3g) but moves to higher altitude. In JJA the region of cold bias near the North Pole broadens in IFV380 (Fig. 11e) compared with the IFV model (Fig. 3h). IFV380 also intensifies the magnitude of cold bias in this region. For ANN, IFV380 intensifies the magnitude of cold bias near both poles and also moves the region with cold bias slightly higher (see Figs. 11f and 3i).
When we further lift the isentropic zone to above 500 K (IFV500), it gives the worst performance of all modeling configurations presented in this paper (Figs. 11g,h,i). In this configuration, the benefit of an isentropic coordinate is only felt near the model lid, which is far above the region where the cold bias is present.
It is interesting to note that moistening in the upper troposphere and lower stratosphere (UTLS) in low latitudes is found in both IFV380 and IFV500 (Figs. 8c,e), which is also present in the IFV core. The moistening effect is almost exactly compensated by the temperature effect in IFV380 (Fig. 8d), which results in very little change in relative humidity and cloud distribution in this region (see Figs. 7b and 9b). The IFV500 core, however, generates substantial cold bias above 100 hPa in low latitudes (Fig. 11i) and thus further increases relative humidity in this region (see Figs. 8f and 7c), which leads to enhancement of cloud fraction (Fig. 10c).
5. Discussion and summary
The main purposes of this study are to describe the implementation of a hybrid-isentropic coordinate within the FV dynamical core and to demonstrate its effectiveness in improving the simulated climate. As presented in this paper, the IFV model was not only capable of realistically simulating climate, but also showed some major improvements over the EUL and FV dynamical cores of CAM. These results revealed that the simulated climate state may be sensitive to the choice of vertical discretization in a GCM.
The most distinct improvement made by the IFV model was the more realistic simulated atmospheric temperatures in the upper troposphere and lower stratosphere at high latitudes. In this study, we demonstrated that the IFV model could effectively reduce the cold pole problem. Reduced simulated cold bias was also obtained in several other models that employed a hybrid-isentropic coordinate (Zhu and Schneider 1997; Webster et al. 1999; Schaack et al. 2004). These numerical studies confirmed a theoretical study by Johnson (1997) in which he argued that an isentropic coordinate system must be used to eliminate the general coldness in GCMs.
The advantage of using an isentrope-based coordinate, for example, the IFV core, over a pressure-based coordinate, for example, the FV core, can be illustrated by a simple two-layer model. During the dynamical time step, the interface between the two layers may be perturbed due to adiabatic dynamics (Fig. 12b). If a pressure-based coordinate is chosen, nonphysical vertical transport can be induced in the vertical remapping stage (Fig. 12c) since some concentration of tracer, one half, is now found in the lower/upper layer. In addition to the nonphysical tracer transport, undesirable vertical heat transport associated with the process is also introduced. Nevertheless, if we use the material surface separating the two fluids as the vertical coordinate, vertical remapping will not introduce any exchange between the two layers. Hence, in climate simulations the air mass within two isentropes may leak through the isentropes during the remapping stage in the FV core while the IFV core can better preserve the air mass within the isentropic layer.
A schematic showing three stages during a dynamical time step in the FV core for a simple two-layer model at (a) initial stage, (b) intermediate stage in which internal gravity waves are excited, and (c) remapping stage in which the vertical coordinate is redefined.
Citation: Journal of Climate 25, 8; 10.1175/2011JCLI4184.1
In Fig. 13, the absolute change of pressure at various model interfaces during the vertical remapping procedure at the end of the dynamical core is plotted. As illustrated in Fig. 13a, the IFV core performs very well in separating the air mass between the isentropic zone and the transitional zone in mid and high latitudes, but it introduces nonphysical vertical transport in the tropics. Nevertheless, the vertical transport in the FV core at similar altitude (θ = 336 K, p ~ 450 hPa in low latitudes and p ~200 hPa in high latitudes, see Fig. 1a) is much larger in the FV core. The larger mass fluxes across model interfaces in the FV core also imply stronger, but mostly undesirable, vertical heat transport.
Absolute change of pressure (hPa) of 1-month average per time step (30 min) at (a) ξ = 336 K for the IFV core, (b) p = 208.1494 hPa for the FV core, and (c) p = 469.0718 hPa for the FV core.
Citation: Journal of Climate 25, 8; 10.1175/2011JCLI4184.1
We have also performed three experiments to compliment the results carried out by the IFV model. As indicated by FVL35, increasing vertical resolution in a pressure-based coordinate achieves little in reducing cold bias while it significantly increases the computational cost. This implies that it is necessary to adequately represent heat transport in the atmosphere, which can be achieved by an isentrope-based coordinate system but not in a pressure-based coordinate, in order to reduce the cold pole problem. The results by IFV380 and IFV500 indicate that it is highly beneficial to set the isentrope defining the isentropic zone as low as possible, but not below the highest potential temperature at the earth’s surface, so that a broader computational domain will enjoy the benefits of isentropic modeling. Among all modeling configurations presented in this paper, the IFV model clearly is the most attractive choice since it brings the most significant improvements but does not add much extra computation cost to the FV core.
It is conceivable that the model coordinate system is highly influential on the simulated tracer transport processes. As revealed from the simulated water vapor (specific humidity) field, the FV and IFV models exhibited several different characteristics. One expected advantage of using an isentropic model was its capability to achieve high accuracy in terms of vertical transport since it was able to eliminate nonphysical vertical transport induced by discrete numerics. Indeed, the IFV core reduced specific humidity up to 60% in mid and low latitudes in the upper troposphere and lower stratosphere (Fig. 6d) compared with the FV model. Nevertheless, an increase in specific humidity of up to 30% was found in the IFV core in the equatorial region at the same altitude where various physical processes, represented by parameterizations, dominated that induced by adiabatic atmospheric flows. Even though the two model coordinates were very similar in the lowest troposphere (both terrain following), a major difference was present in the lowest equatorial troposphere: more water vapor was present in this region in the IFV model.
Independent of the dynamical core employed, the NCAR CAM has been known to overestimate the fraction of high clouds. Since the IFV model simulated a warmer upper troposphere/lower stratosphere, it led to a much lower relative humidity in these regions. Consequently, the formation of high clouds is highly suppressed. Thus, a substantial reduction in the amount of high clouds was observed in the IFV model. This also marks a significant improvement achieved by the IFV model.
Even though it was evident that the IFV made several major improvements in its simulated climate, some degradation was also introduced. The biggest degradation by the IFV model was the creation of cold temperature bias in the equatorial stratosphere, which was much weaker in the FV model. Several factors may contribute to this degradation. First, the hybrid-isentropic coordinate employed in this study possessed a very coarse resolution near the model top where state variables were poorly resolved. This might lead to large errors in these regions during a model integration. Second, the NCAR CCM3 physics suite was designed and tuned for a σ − p coordinate system, which might not provide an optimal performance within a σ − θ coordinate framework. It is possible that the large bias in the IFV model can be profitably decreased by carefully retuning the model.
Acknowledgments
The authors are indebted to Henry Tufo, Ram Nair, Peter Lauritzen, Joe Tribbia, and Don Johnson for helpful discussions. We would also like to thank three reviewers for their input on the work and S. J. Lin for conversations during the development of the paper. The work is partially funded by the National Center for Atmospheric Research through the National Science Foundation, DOE SiDAC program under Award DE-FG02-04ER63870 and the DOE Petascale program under Award DE-FG-2-07ER64464.
APPENDIX
The Positive Definite Limiter
During a simulation by the IFV model, the semi-Lagrangian flux-form transport scheme (described in Lin and Rood 1996) occasionally generates negative mass, typically found within the lowest isentropic domain, which leads to failure of a numerical integration. A similar scenario was also reported in Schaack et al. (2004). To ensure that a simulation advances successfully without producing negative mass, which in turn causes negative potential temperature and tracer mixing ratios, we implemented a positive definite limiter in the transport scheme, which is proposed by Skamarock (2006). This limiter is applied to the integration of the mass continuity equation by correcting the mass fluxes across cell walls of all grid points. This is found to be sufficient to keep the mass field, potential temperature field, and all tracers positive.
In addition to implementing a positive definite limiter in the transport scheme, another step is taken to reduce the possibility of triggering the limiter during a simulation. This helps ensure that the mass fluxes are not constantly adjusted so that a higher order of accuracy in transport is preserved. As shown in Fig. 1, the mass within an isentropic layer can become extremely small, especially in the lower isentropic zone near the poles. This implies that the meridional gradients in the mass field could be very large at high latitudes, which could potentially impose a severe challenge on the transport scheme to accurately predict the fluxes. To remove some of the sharp gradients we follow the strategy described in Schaack et al. (2004) and constrain the mass at any grid point so that it cannot fall under 1.5 hPa. During the vertical remapping stage, the PPM algorithm is employed to construct the subgrid vertical distribution of potential temperature. Since the PPM algorithm allows discontinuity at model interfaces, it could lead to massless layers occasionally in the isentropic zone. If the subgrid distribution of potential temperature indicates that some grid points should possess mass less than the threshold, the upper-model interface at those points is lifted to accommodate this criterion. This threshold was chosen in Schaack et al. (2004) and it is effective in removing massless layers and minimizing the triggering of the positive definite limiter in our simulations.
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