Improving energetics in an ideal baroclinic instability case with a Physical Conserving Fidelity model

Abstract

To improve the energetics in the life cycle of an ideal baroclinic instability case, we develop a Physical Conserving Fidelity model (F-model), and we compare the simulations from the F-model to those of the traditional global spectral semi-implicit model (control model). The results for spectral kinetic energy and its budget indicate different performances at smaller scales in the two models. A two-way energy flow emerges in the generation and rapid growth stage of the baroclinic disturbance in the F-model. However, only a downscale mechanism dominates in the control model. In the F-model, the meso- and smaller scales are energized initially, and then an active upscale nonlinear cascade occurs. Thus, disturbances at prior scales are forced by both downscale and upscale energy cascades and by conversion from potential energy. An analysis of the eddy kinetic energy budget also shows remarkable enhancement of the energy conversion rate in the F-model. As a result, characteristics of the ideal baroclinic wave are greatly improved in the F-model, in terms of both intensity and time of formation.

Keywords

energy conversion / energy cascade / ideal baroclinic instability / high order total energy conservation / time-split scheme

Introduction

Fundamental predictability studies (e.g., Lorenz, 1963; Leith and Kraichnan, 1972) have shown that the two-way cascade of energy between slowly evolving large-scale circulations and smaller-scale, resolved and unresolved motions are a crucial factor in the predictability of chaotic dynamical systems. Several recent studies have focused on the up-scale cascade of energy, which is understood as a critical element in increasing forecasting abilities. The role of up-scale energy cascades in the life cycles of synoptic-scale weather systems has received much attention, both in theory and in practical applications (e.g., The Observing System Research and Predictability Experiment program). Baroclinic instability is one of the most basic and important mechanisms of atmospheric dynamics (Charney, 1947; Eady, 1949). Much attention has been focused on the ability of numerical models to describe the development of baroclinic waves (e.g., Hoskins and Simmons, 1975; Giraldo and Rosmond, 2004; Polvani et al., 2004; Jablonowski and Williamson, 2006). These studies have shown that the development of baroclinic waves shows weaker patterns at lower resolutions, and better performance may be achieved by increasing the resolution of numerical models (Jablonowski and Williamson, 2006). This ideal baroclinic wave test is currently in wide use by the numeric modeling community (e.g., Williamson et al. 2009; Lauritzen et al., 2010; Shields et al., 2012; Skamarock et al., 2012; Ullrich and Jablonowski, 2012a,2012b). However, an increasing number of studies in the numeric modeling community have understood the importance of correctly describing the non-linear interactions between scales - particularly improving numerical and physical representation of the upscale energy cascade from meso- and smaller scales (e.g., Skamarock, 2004; Vallgren et al., 2011; Bierdel et al., 2012; Ngan and Eperon, 2012; Talbot et al., 2012). In this study, a physical conserving fidelity scheme that constantly possesses a cubic high-order total-energy-conserving global integral property in both spatial and temporal discretizations is applied to an ideal baroclinic instability case. The energy flow and energy conversion rate during the lifecycle of the baroclinic wave are investigated. The results are directly compared with those from a traditional semi-implicit scheme.

Methods

The traditional global spectral semi-implicit model for baroclinic primitive equations is commonly used in many countries throughout the world for operational weather forecasting and numerical simulations of general circulation (Bourke, 1974; Ji et al., 1990; Zhang et al., 1995; Collins et al., 2004), this model is taken as the control model in this study (Ji et al., 1990; Zhang et al., 1995). However, maintaining high-order total energy conservation in the time-split has not yet been solved in the control model over a long period of time. In this paper, we introduce a Physical Conserving Fidelity model (F-model) that conserves total energy to high order in both explicit and semi-implicit time-split formulations. By retaining corresponding integral conserving properties in both space and time discretizations, the problems involving linear and nonlinear computational instability are solved. Thus, the F-Model is able to run without employing any artificial diffusive techniques (Zhong, 1999) that are typically necessary to stabilize the Eulerian model.

The mathematical theorem

The operator equation of the evolution problem may be stated as the following:

(1){\displaystyle \frac{\partial u}{\partial t}}+Au=\mathrm{0.}

Here, u denotes physical variables, t is time, and A is formulation operator.

Based on a general compensation principle and the inverse formulation method of a fidelity scheme (Zhong, 1992), a general physical conservation law for the time-difference fidelity scheme of Eq. (1) may be written as the following:

(2){\displaystyle \frac{u^{n+\mathrm{1}}-u^n}{\mathrm{\Delta }t}}+(A^n-A_L^n)u^n+A_L^n{u}^{n+\mathrm{1}}+\mathrm{\Delta }t{ϵ}^n{B}^n{u}^n=\mathrm{0.}

Here, A_L is an auxiliary formulation operator, {ϵ}^n is an undetermined compensation coefficient and Bⁿuⁿ is a compensation operator.

In particular, if operator Eq. (1) yields a general cubic conserving integral property,

(3){\displaystyle \int (A_{\mathrm{1}}u\cdot A_{\mathrm{2}}u\cdot A_{\mathrm{3}}Au+A_{\mathrm{1}}Au\cdot A_{\mathrm{2}}u\cdot A_{\mathrm{3}}u+A_{\mathrm{1}}u\cdot A_{\mathrm{2}}Au\cdot A_{\mathrm{3}}u)}d\sigma =\mathrm{0},

then the following theorem holds true. Here, A₁, A₂, and A₃ are all bounded-space operators that are independent of u and t; d\sigma is a space integral element.

Theorem: suppose the compensation coefficient {ϵ}^n satisfies

(4)\mathrm{\Delta }{t}^{\mathrm{4}}{\gamma }_{\mathrm{1}}{({ϵ}^n)}^{\mathrm{3}}+\mathrm{\Delta }{t}^{\mathrm{2}}{\gamma }_{\mathrm{2}}{({ϵ}^n)}^{\mathrm{2}}+{\gamma }_{\mathrm{3}}{ϵ}^n+{\gamma }_{\mathrm{4}}=\mathrm{0}

and its order of magnitude is O(\mathrm{\Delta }t⁰); then scheme (2) is a fidelity scheme with a cubic conserving integral property and is compatible with Eq. (1), where

(5.1){\gamma }_{\mathrm{1}}={\displaystyle \int A_{\mathrm{1}}}L{B}^n{u}^n\cdot A_{\mathrm{2}}L{B}^n{u}^n\cdot A_{\mathrm{3}}L{B}^n{u}^{n}d\sigma

(5.2){\gamma }_{\mathrm{2}}=-{\displaystyle \int (A_{\mathrm{1}}L{B}^n{u}^n\cdot A_{\mathrm{2}}L{B}^n{u}^n\cdot A_{\mathrm{3}}M{u}^n+A_{\mathrm{1}}L{B}^n{u}^n\cdot A_{\mathrm{2}}M{u}^n\cdot A_{\mathrm{3}}L{B}^n{u}^n+A_{\mathrm{1}}M{u}^{n\cdot }\cdot A_{\mathrm{2}}L{B}^n{u}^n\cdot A_{\mathrm{3}}L{B}^n{u}^n)}d\sigma

(5.3){\gamma }_{\mathrm{3}}={\displaystyle \int (A_{\mathrm{1}}L{B}^n{u}^n\cdot A_{\mathrm{2}}M{u}^n\cdot A_{\mathrm{3}}M{u}^n+A_{\mathrm{1}}M{u}^n\cdot A_{\mathrm{2}}L{B}^n{u}^n\cdot A_{\mathrm{3}}M{u}^n+A_{\mathrm{1}}M{u}^{n\cdot }\cdot A_{\mathrm{2}}M{u}^n\cdot A_{\mathrm{3}}L{B}^n{u}^n)}d\sigma

(5.4){\gamma }_{\mathrm{4}}=-{\displaystyle \int (A_{\mathrm{1}}K{u}^n\cdot A_{\mathrm{2}}{A}^n{u}^n\cdot A_{\mathrm{3}}{A}^n{u}^n+A_{\mathrm{1}}{A}^n{u}^n\cdot A_{\mathrm{2}}{u}^n\cdot A_{\mathrm{3}}{A}^n{u}^n+A_{\mathrm{1}}{A}^n{u}^n\cdot A_{\mathrm{2}}{A}^n{u}^n\cdot A_{\mathrm{3}}{u}^n+A_{\mathrm{1}}L{A}_L^n{A}^n{u}^n\cdot A_{\mathrm{2}}M{u}^n\cdot A_{\mathrm{3}}M{u}^n+A_{\mathrm{1}}K{u}^n\cdot A_{\mathrm{2}}M{u}^n\cdot A_{\mathrm{3}}L{A}_L^n{A}^n{u}^n+A_{\mathrm{1}}K{u}^n\cdot A_{\mathrm{2}}L{A}_L^n{A}^n{u}^n\cdot A_{\mathrm{3}}K{u}^n)}d\sigma

where K=I-\mathrm{\Delta }t{A}^n, M=I-\mathrm{\Delta }t{A}^n+\mathrm{\Delta }{t}^{\mathrm{2}}L{A}_L^n{A}^n, L is an inverse operator of I+\mathrm{\Delta }t{A}_L^n, and I is a unit operator.

Based on this formulation theorem, a total-energy-conserving semi-implicit time-difference scheme for global spectral-vertical finite-difference models of baroclinic primitive equations is formulated and realized.

The numerical method of the F-Model is described in great detail in Zhong (1999). Vorticity, divergence, temperature, and surface pressure are represented horizontally by a truncated triangular series of spherical harmonics. The model utilized an Eulerian treatment of the full primitive equations with a two-time-level semi-implicit time discretization. In this paper, we adopt T42L9 as the space resolution. The time step is 30 minutes. All results are run without a horizontal diffusion scheme, which avoids artificial smoothing at truncation scales (to a certain extent).

Experimental design

The baroclinic instability test case suggested by Jablonowski and Williamson (2006) will be applied to the dynamical core of the control model and the F-model in this paper. The two dynamical cores are initialized with steady-state, balanced initial conditions, which guarantee static, inertial, and symmetric stability properties but involves the baroclinic and barotropic instability mechanisms. Then a relatively large-scale but localized Gaussian hill perturbation is superimposed onto the zonal wind in the northern mid-latitudes, which triggers the evolution of the baroclinic wave over the course of several days.

Results

Evolution of the ideal baroclinic wave

Jablonowski and Williamson (2006) have applied the aforementioned baroclinic instability test on a variety of model formulations and grids with various resolutions. When the resolution equals T170, the lifecycle of the baroclinic wave tends to be convergent. This reference solution suggests that the wave generally starts growing observably near day 4 and evolves rapidly and sharply thereafter (particularly after day 7); cyclogenesis is explosive at day 8, and the wave train breaks after day 9.

Figures 1 and 2 show a time sequence of the evolving surface pressure and 850 hPa temperature fields of the two models; we can see that both the control model and the F-model can successfully simulate the entire lifecycle of a baroclinic wave, including the initial generation, rapid development, and the final wave breaking. However, there are significant differences in the intensity of the vortex and time lag of formation between the two models.

First, the baroclinic wave in the control model is generated two days later than in the F-model. In the F-model, the wave begins to grow on day 4 with a small-amplitude disturbance. On day 6, the surface pressure shows two weak high- and low-pressure systems (Fig. 1(f)) and two visible waves in the temperature field (Fig. 2(f)). By day 8, the highs and lows have deepened significantly (Fig. 1(g)); the two waves have almost peaked and are beginning to wrap around with fronts trailing from them, and a third upstream wave is now visible (Fig. 2(g) ). By day 10, wave breaking has set in, and there are three closed cells with the leading front being quite sharp.

Second, the intensity of the cyclone and the vorticity gradient appear enhanced in the F-model (Fig. 3). Figure 3 presents the evolution of the maximum value of the vorticity and the vorticity gradient at 850 hPa. In the control model, the baroclinic perturbation starts on day 6, whereas the increase of vorticity occurs on day 4 in the F-model. In spite of the time lag, the control model reaches its maximum vorticity value of approximately 20e-5 s^-1 (Fig. 3(a)) at maturity (on day 10), whereas the value increases to 24e-5 s^-1 in the F-model (on day 8). Similar trends may be observed in the vorticity gradient (Fig. 3(b)) that represents the smaller scale steep-gradient features.

Spectral kinetic energy (KE)

From the perspective of energetics, the development of a synoptic system always accompanies accumulation of KE at certain prior scales. To investigate the energy flow between scales, the spectral distribution of KE throughout the lifecycle of the baroclinic wave is examined. Figures 4 (a) and 4 (c) present the spectral and time distributions of the KE spectral density in the control and F-model, respectively. A concentration of energy occurs at the prior scales (longitudinal waves number 10 to 15). In accordance with the development of the baroclinic wave, the energy concentration in the control model at approximately day 8 is maintained until dissipation near day 11 (Fig. 4(a)). In the F-model, the energy congregates at the prior scales on day 6-which corresponds to the genesis of the baroclinic wave-and dissipates near day 9. Another notable difference is the energy distribution at smaller scales (from wave number 25 to 42). The energy density at those smaller scales in the F-model is far greater than in the control model at an early stage of development.

To display the energy flow between scales with time, we subtract the energy spectral density along the time series (shown in Figs. 4 (b) and 4(d)). The shaded regions represent the decrease of energy compared to the previous day, which means that energy has decreased and moved to other spectral elements. It is notable that a clear image of a two-way energy sheer in the F-model (Fig. 4(d)) has emerged, whereas there is only one-way flow (from larger scales to the prior scales) in the control model (Fig. 4(b)).

In the F-model (Figs. 4(c) and 4(d)), the prior scales (waves number 10 to 15) obtain continued energy from day 4 onward; both the smaller scales (waves 16 to 42) and the larger scales (waves 5 to 10) are losing energy contemporaneously. At maturity (days 7 to 8) - which corresponds to the rapid growth of the baroclinic cyclone - the identical pattern of energy transformation continues. Moreover, energy has similarly increased at smaller scales, such as at wave numbers near 23 and 28, which is consistent with the role of small disturbances in the development of baroclinic instability. This may be one of the reasons that the meso-scale gradients are better reflected in the F-model. In the wave-breaking stage (day 8), the energy at the prior scales starts to dissipate and energy grows at both smaller and larger scales simultaneously. However, in the control model (Figs. 4(a) and 4(b)), the energy near the truncation scales is rather weak on day 4, and a process of gathering energy on these scales is visible. The increment on smaller scales slows down until day 6, and energy at the larger scales continues to decrease (wave numbers 5 to 10). The downscale flows dominate from day 6 to day 9, when the baroclinic wave grows to maturity (from day 9 onward).

Thus, we observe that the energy transformation from smaller scales plays a significant role for the F-model in describing the life cycle of a baroclinic wave, including enhancing the intensity of the vortex and improving the time of formation.

Spectral kinetic energy budget

During the growth of the baroclinic wave, we can detect and quantify the contribution of other scales to the prior scales (wave numbers 10 to 15) by computing the spectral KE budgets. This requires knowledge of the contribution of each term in the prognostic model equation for KE at each wave number. For the sake of simplicity in interpreting results, a pressure coordinate system is used. The equation for the rate of change of KE in the pressure coordinate system is the following (Smagorinsky et al., 1965):

(6){\displaystyle \frac{\partial K}{\partial t}}=-\nabla \cdot \left(\overrightarrow{V}K\right)-{\displaystyle \frac{\partial \omega K}{\partial P}}-\overrightarrow{V}\cdot \nabla \varphi +\overrightarrow{V}\cdot F,

where V is earth velocity, F is frictional force (equal to zero in this study), and KE K=V²/2. The terms on the right-hand side are the advection term, the pressure interaction term, the source term and the sink term of KE, respectively. The spectral KE budget is obtained by calculating each term on the right-hand side of Eq. (6) on the Gaussian grid and spectrally transforming it to obtain spherical harmonic coefficients for each field (Koshyk and Hamilton, 2001).

The results show that conversion to/from potential energy and nonlinear interactions among scales are two significant mechanisms in the KE budget. Figure 5 shows the spectral KE budgets of these two terms as a function of the wave number (n) during the development of the ideal baroclinic wave. There are obvious differences in both structure and intensity in the two models.

In the control model, the effect of the nonlinear advection terms in Fig. 5(a) is mainly to remove KE from larger scales (larger than wave number 23) and add KE to smaller scales before day 6. Both the removal of KE from larger scales and the increment of KE to smaller scales are consistent with a downscale KE cascade. At larger wave numbers, only a small transfer occurs when the baroclinic waves begin to grow. However, the F-model (Fig. 5(b)) is different, in which the advection contribution to KE from the meso-scales is quite substantial. In the first stage (before day 4), the advection term acts to redistribute KE among the scales. Large scales are mainly loose KE with a downscale cascade. Scales at wave numbers 28 to 35 disperse energy until day 5 and display an upscale and downscale mechanism. The middle scales (wave numbers 14 to 28) and the smallest scales (wave numbers larger than 35) are obtaining energy. With the development of the baroclinic wave, energy is removed from those smallest and middle scales. At the stage of rapid growth, we clearly observe an accumulation of energy at the prior scales (wave number 10 to 15, from day 5 to 8). This suggests that the internal nonlinear horizontal cascade energizes the meso- and small scales at first; then the nonlinear cascade from these smaller and larger scales flows to the prior scales during the growth of the baroclinic wave, which represents a picture of a two-way energy flow.

A prominent KE source is the pressure gradient term, which is shown in Figs. 5(c) and 5(d) and represents the internal conversion of potential energy to KE. The spectra field of this term is not particularly smooth, but it also presents an interesting result in the F-model (Fig. 5(d)). The distribution of KE change at larger scales (n<18) is fairly regular. During the first stage (before day 4), the conversion term is positive for most middle and smaller scales (n>20), which indicates a conversion from potential energy to KE. When the baroclinic wave begins to grow, positive conversion occurs at meso-scales (wave numbers 18 to 25). At the stage of rapid growth, the centers of positive maximum may be detected in the prior scales (wave numbers 10 to 15, from days 5 to 8). Figs. 5 (b) and 5(d) represent the relative contribution of the two mechanisms. The increment of KE in the prior scales caused by transformation from a nonlinear cascade is clear and continual, whereas in the conversion term, it is not as integrated but has several positive centers. More continuous conversion may be observed in the meso-scales (wave numbers 20 to 25). It also shows that the values of positive contribution from the conversion term are larger than those of the advection term.

This study presents a fairly different picture of energetics during the growth of an ideal baroclinic wave between the two models. In the F-model, the meso and smaller scales are more energized initially, and the upscale nonlinear cascade is much more active. Waves of prior scales are then forced by both downscale and upscale energy cascade. Another significant difference between the F-model and control model is that the quantum of the corresponding terms is much larger in the former. To investigate the effects on the baroclinic disturbance, the terms of the eddy KE budget are analyzed.

Eddy kinetic energy budget

The equations for the rate of change of the hemispheric mean of eddy KE may be written as follows:

(7)

where,

(8)K_E={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}[{(u\prime )}^{\mathrm{2}}+{(v\prime )}^{\mathrm{2}}];K_Z={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\left[{\left({\overline{u}}^{\lambda }\right)}^{\mathrm{2}}+{\left({\overline{v}}^{\lambda }\right)}^{\mathrm{2}}\right],

(9)

(10)

where the terms on the right side of Eq.(7) represent the vertical advection term, the eddy conversion term, the eddy pressure interaction term, the eddy dissipation, the remaining part of the transfer from K_z to K_E, and the energy transfer from zonal KE into the eddy KE attributed to the vertical interaction term (Reynolds stress by the large-scale eddies).

The results show a synchronization change of the growth of the baroclinic disturbance and the increase of eddy KE. In the control model, the transform occurs and grows approximately two days later and much weaker than in the F-model. Ignoring problems of time lag, the terms of the eddy KE budget at the stage of rapid growth is shown in Fig. 6. In both models, the conversion term from potential energy to eddy KE is largest. The vertical distribution and direction of energy transfer are shown to be consistent in the two models. However, the magnitude in the F-model is larger on a 1-2 order than that in the control model, which indicates that the F-model can (remarkably) enhance the energy conversion rate and obtain more energy. This is one of the reasons that the strength of the vortex is greatly enhanced in the F-model.

Discussion

In this paper, a Physical Conserving Fidelity model (F-model) was used to study the energetics throughout the life cycle of an ideal baroclinic instability case. We compared the simulations with those of the traditional global spectral semi-implicit model (the control model). An evident two-way energy flow emerges in the F-model. By contrast, only a downscale mechanism dominates in the control model. The study on spectral KE budget further presents differences in the energetics between the two models in terms of both structure and intensity. In the F-model, the meso- and smaller scales are more energized at first and then the upscale nonlinear cascade is much more active. The analysis of the eddy KE budget also shows a remarkable enhancement of the energy conversion rate in the F-model. This study demonstrates that the baroclinic disturbances at prior wavelengths are mostly forced by both downscale and upscale energy cascades and by conversion from potential energy in the F-model. As a result, the intensity and the time of formation of the baroclinic wave are greatly improved.

Physical conservation laws in the time-split scheme are realized in the F-model but have not been resolved in the control model. The modeling community generally views semi-implicit schemes as an acceptable trade of small-scale accuracy for global efficiency. However, this study indicates that maintenance of physical conservation laws in the temporal discretization has significant effects on the short-term development of the ideal baroclinic instability. The formation and intensity of the baroclinic wave are greatly improved. Most notably, a two-way energy cascade is discerned in the F-model; the upscale energy flow is more active, and the energy conversion rate is greatly enhanced. Thus, the maintenance of physical conserving laws in the numerical discretization is both a constraint and a critical factor in enhancing energy conversion rates and improving energy cascades between scales.

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