Global diagnostic energetics of five state-of-the-art climate models

Abstract

In this study, the global energy cycle of five state-of-the-art climate models is evaluated in the wave number domain for all seasons. The energy cycle estimates are based on 30 years of 6-hourly data obtained at pressure levels of all models. The models energetics are compared to those obtained from three reanalysis datasets (ERA-40, JRA-25 and NCEP-R2). The results show that the distributions of the energetics integrands and the shape of the various wave number spectra are reasonably well simulated. Many important features can be found in most models, namely both the upscale and downscale energy cascade for the wave–wave interactions of kinetic energy, the downscale energy cascade for the wave–wave interactions of available potential energy and the downscale energy transfer for the zonal–wave interactions of kinetic energy. However, the magnitude in the integrands distributions is generally excessive, yielding too much energy and an overactive energy cycle in the models. Accordingly, this energy excess is also reflected in the various spectra, specially but not exclusively, at the synoptic scale wave numbers for the energy conversion/transfer rates. The well known cold pole bias and the too strong tropospheric jets, along with their dislocation in some cases, still persist in the climate models. These are some of the deficiencies in the models directly implicated in the energy cycle. Apparently, simply increasing the horizontal and vertical resolutions is not enough to eliminate these deficiencies due to somewhat opposite effects achieved by refining both spatial resolutions. Therefore, more accurate physics parameterisations as well as improved numerical schemes and resolution dependence of parameterisations seem to be essential for a significant improvement in the models energetics. Moreover, efforts should be made to improve the physical processes controlling the generation of zonal available potential energy and dissipation of eddy kinetic energy, in which the synoptic scale should be fundamental, as inferred from the excessive energy conversion/transfer rates in the models spectra.

Appendix

1.1 Operations on an arbitrary function f

• Time average, $\bar{f}:$

  $$ \bar{f} = {\frac{1} {t_2 - t_1}} \int\limits_{t_1}^{t_2} f \hbox{d}t, $$

• Zonal average, [f]:

  $$ \left[f\right] = {\frac{1} {2 \pi}} \int\limits_{0}^{2 \pi} f \hbox{d}\lambda, $$

• Departure from zonal average, f ^*:

  $$ f^{*} = f-\left[f\right] $$

• Area average over an isobaric surface, $\tilde{f}:$

  $$\tilde{f} = {\frac{1}{{4\pi }}}\int\limits_{0}^{{2\pi }} {\int\limits_{{ - {\frac{\pi }{2}}}}^{{{\frac{\pi }{2}}}} {f\,\cos \phi \,{\text{d}}\phi \,{\text{d}}\lambda } }$$

• Departure from area average, f′′:

  $$ f^{\prime\prime} = f-\tilde{f} $$

Fourier transform pair for harmonic analysis around latitude circles:

$$ \begin{aligned} f\left(\lambda,\phi,p,t \right) &= \sum_{n=-\infty}^{\infty} F\left(n,\phi,p,t \right) e^{in\lambda}\\ F\left(n,\phi,p,t \right) &= {\frac{1} {2 \pi}}\int\limits_{0}^{2 \pi} f\left(\lambda,\phi,p,t \right) e^{-in\lambda} \hbox{d}\lambda \end{aligned} $$

$$ \begin{array}{llllllllll} f\left(\lambda\right) & : & u & v & \omega & Z & T & h & x & y\\ F\left(n\right) & : & U & V & \Upomega & A & B & H & X & Y\\ \end{array} $$

1.2 Expressions for the quantities in the balance Eqs. (1, 2, 3 and 4)

$$ P\left(0 \right) = \int\limits_{{\mathcal{M}}} \gamma {\frac{c_p} {2}} \left(\left[ T\right]^{\prime\prime}\right)^2 \hbox{d}m $$

(5)

$$ P\left(n \right) = \int\limits_{{\mathcal{M}}} \gamma c_p \vert B(n)\vert^2 \hbox{d}m $$

(6)

$$ K\left(0 \right) = \int\limits_{{\mathcal{M}}} {\frac{1} {2}} \left( \left[ u\right]^2 + \left[ v\right]^2\right) \hbox{d}m $$

(7)

$$ K\left(n \right) = \int\limits_{{\mathcal{M}}} \left( \vert U(n)\vert^2 + \vert V(n)\vert^2\right) \hbox{d}m $$

(8)

$$ R\left(n \right) = \int\limits_{{\mathcal{M}}} \gamma c_p \left( V(n) B(-n) + V(-n) B(n)\right) {\frac{1} {a}}\,{\frac{\partial \left[ T\right] } {\partial \phi}} \hbox{d}m + \int\limits_{{\mathcal{M}}} \gamma c_p \left( \Upomega(n) B(-n) + \Upomega(-n) B(n)\right) {\frac{T} {\Uptheta}}\,{\frac{\partial \left[ \Uptheta\right]^{\prime\prime} } {\partial p}} \hbox{d}m $$

(9)

$$ \begin{aligned} C\left(n \right) &= - \int\limits_{{\mathcal{M}}} {\frac{R} {p}} \left( \Upomega(n) B(-n) + \Upomega(-n) B(n)\right) \hbox{d}m\\ &= - \int\limits_{{\mathcal{M}}} {\frac{i n g} {a \hbox{cos}} \phi} \left( A\left(n \right) U\left(-n \right) - A\left(-n \right) U\left(n \right) \right) \hbox{d}m\\ & + \int\limits_{{\mathcal{M}}} {\frac{g} {a}} \left( {\frac{\partial A(n)} {\partial \phi}} V\left(-n \right) + {\frac{\partial A(-n)} {\partial \phi}} V\left(n \right)\right) \hbox{d}m \end{aligned} $$

(10)

$$ \begin{aligned} M\left(n \right) &= - \int\limits_{{\mathcal{M}}} 2 U(n) U(-n) \left[ v\right] {\frac{\tan \phi} {a}}\,\hbox{d}m\\ & + \int\limits_{{\mathcal{M}}} \left( 2 V(n) V(-n) {\frac{1} {a}}\,{\frac{\partial \left[ v\right] } {\partial \phi}} + \left( V(n) \Upomega(-n) + V(-n) \Upomega(n)\right) {\frac{\partial \left[ v\right] } {\partial p}}\right) \hbox{d}m\\ & + \int\limits_{{\mathcal{M}}} \left( U(n) \Upomega(-n) + U(-n) \Upomega(n)\right) {\frac{\partial \left[ u\right]} {\partial p}} \hbox{d}m\\ & + \int\limits_{{\mathcal{M}}} \left( U(n) V(-n) + U(-n) V(n)\right) {\frac{\cos \phi} {a}}\,{\frac{\partial} {\partial \phi}} \left( {\frac{\left[ u\right]} {\cos \phi}}\right) \hbox{d}m \end{aligned} $$

(11)

$$ \begin{aligned} C\left(0 \right) &= - \int\limits_{{\mathcal{M}}} {\frac{R} {p}} \left[ \omega\right]^{\prime\prime} \left[ T\right]^{\prime\prime} \hbox{d}m\\ &= - \int\limits_{{\mathcal{M}}} {\frac{g} {a}} \left[ v\right] {\frac{\partial \left[ Z\right] } {\partial \phi}} \hbox{d}m \end{aligned} $$

(12)

$$ \begin{aligned} S\left(n \right) &= - \int\limits_{{\mathcal{M}}} \gamma c_p {\sum\limits_{{\mathop{m=-\infty}\limits_{m\neq 0} }}^{\infty}}{\frac{in} {a \cos \phi}} B(m) \left(B(-n) U(n-m) + B(n) U(-n-m)\right) \hbox{d}m\\ &+ \int\limits_{{\mathcal{M}}} \gamma c_p {\sum\limits_{{\mathop{m=-\infty}\limits_{m\neq 0} }}^{\infty}}{{\frac{1} {a}} B(m) \left( V(n-m) {\frac{\partial B(-n)} {\partial \phi}} + V(-n-m) {\frac{\partial B(n)} {\partial \phi}} \right) \hbox{d}m}\\ &+ \int\limits_{{\mathcal{M}}} \gamma c_p {\sum\limits_{{\mathop{m=-\infty}\limits_{m\neq 0} }}^{\infty}}{B(m) \times\left( \Upomega(n-m) {\frac{\partial B(-n)} {\partial p}} + \Upomega(-n-m) {\frac{\partial B(n)} {\partial p}}\right) \hbox{d}m}\\ &+ \int\limits_{{\mathcal{M}}} \gamma c_p {\sum\limits_{{\mathop{m=-\infty}\limits_{m\neq 0} }}^{\infty}}{{\frac{R} {p c_p}} B(m) \left(\Upomega(n-m) B(-n) + \Upomega(-n-m) B(n) \right) \hbox{d}m} \end{aligned} $$

(13)

$$ \begin{aligned} L\left(n \right) &= - \int\limits_{{\mathcal{M}}} {\sum\limits_{{\mathop{m=-\infty}\limits_{m\neq 0} }}^{\infty}} {\frac{i n} {a \cos \phi}} U\left( m\right) \left( U\left( -n\right) U\left( n-m\right) - U\left( n\right) U\left( -n-m\right) \right) \hbox{d}m\\ &+ \int\limits_{{\mathcal{M}}} {\sum\limits_{{\mathop{m=-\infty}\limits_{m\neq 0} }}^{\infty}}{{\frac{i n} {a \cos \phi}} V\left( m\right) \left( V\left( -n\right) U\left( n-m\right) - V\left( n\right) U\left( -n-m\right) \right) \hbox{d}m}\\ &- \int\limits_{{\mathcal{M}}} {\sum\limits_{{\mathop{m=-\infty}\limits_{m\neq 0} }}^{\infty}}{{\frac{\tan \phi} {a}} V\left( m\right) \left( U\left( n-m\right) U\left( -n\right) + U\left( -n-m\right) U\left( n\right) \right) \hbox{d}m}\\ &+ \int\limits_{{\mathcal{M}}} {\sum\limits_{{\mathop{m=-\infty}\limits_{m\neq 0} }}^{\infty}}{{\frac{\tan \phi} {a}} U\left( m\right) \left( U\left( n-m\right) V\left( -n\right) + U\left( -n-m\right) V\left( n\right) \right) \hbox{d}m}\\ &+ \int\limits_{{\mathcal{M}}} {\sum\limits_{{\mathop{m=-\infty}\limits_{m\neq 0} }}^{\infty}}\left( U(-n){\frac{\partial} {\partial p}} \left(U(m) \Upomega(n-m)\right) + U(n) {\frac{\partial} {\partial p}} \left(U(m) \Upomega(-n-m)\right)\right) \hbox{d}m\\ &+ \int\limits_{{\mathcal{M}}} {\sum\limits_{{\mathop{m=-\infty}\limits_{m\neq 0} }}^{\infty}}\left( V(-n) {\frac{\partial} {\partial p}} \left(V(m) \Upomega(n-m)\right) + V(n) {\frac{\partial} {\partial p}} \left(V(m) \Upomega(-n-m)\right)\right)\hbox{d}m\\ &+ \int\limits_{{\mathcal{M}}} {\sum\limits_{{\mathop{m=-\infty}\limits_{m\neq 0} }}^{\infty}}{{\frac{1} {a \cos \phi}} U\left( -n\right) {\frac{\partial} {\partial \phi}} \left( U\left( m\right) V\left( n-m\right) \cos \phi \right) \hbox{d}m}\\ &+ \int\limits_{{\mathcal{M}}} {\sum\limits_{{\mathop{m=-\infty}\limits_{m\neq 0} }}^{\infty}}{{\frac{1} {a \cos \phi}} U\left( n\right) {\frac{\partial} {\partial \phi}} \left( U\left( m\right) V\left( -n-m\right) \cos \phi \right) \hbox{d}m}\\ &+ \int\limits_{{\mathcal{M}}} {\sum\limits_{{\mathop{m=-\infty}\limits_{m\neq 0} }}^{\infty}}{{\frac{1} {a \cos \phi}} V\left( -n\right) {\frac{\partial} {\partial \phi}} \left( V\left( m\right) V\left( n-m\right) \cos \phi \right) \hbox{d}m}\\ &+ \int\limits_{{\mathcal{M}}} {\sum\limits_{{\mathop{m=-\infty}\limits_{m\neq 0} }}^{\infty}}{{\frac{1} {a \cos \phi}} V\left( n\right) {\frac{\partial} {\partial \phi}} \left( V\left( m\right) V\left( -n-m\right) \cos \phi \right) \hbox{d}m}\\ \end{aligned} $$

(14)

$$ G\left(0 \right) = \int\limits_{{\mathcal{M}}} \gamma \left[ T\right]^{\prime\prime} \left[ h\right]^{\prime\prime} \hbox{d}m $$

(15)

$$ G\left(n \right) = \int\limits_{{\mathcal{M}}} \gamma \left( B(n) H(-n) + B(-n) H(n)\right) \hbox{d}m $$

(16)

$$ D\left(0 \right) = - \int\limits_{{\mathcal{M}}} \left( \left[ u\right] \left[ x\right] + \left[ v\right] \left[ y\right]\right) \hbox{d}m $$

(17)

$$ \begin{aligned} D\left(n \right) &= - \int\limits_{{\mathcal{M}}} \left( U\left( n\right) X\left( -n\right) + U\left( -n\right) X\left( n\right) \right) \hbox{d}m\\ & - \int\limits_{{\mathcal{M}}} \left( V\left( n\right) Y\left( -n\right) + V\left( -n\right) Y\left( n\right) \right) \hbox{d}m \end{aligned} $$

(18)

$$ \gamma = - {\frac{\theta} {T}} {\frac{R} {p C_p}} \left({\frac{\partial \tilde{\theta}} {\partial p}}\right)^{-1} $$

(19)