The LMDZ4 general circulation model: climate performance and sensitivity to parametrized physics with emphasis on tropical convection

Abstract

The LMDZ4 general circulation model is the atmospheric component of the IPSL–CM4 coupled model which has been used to perform climate change simulations for the 4th IPCC assessment report. The main aspects of the model climatology (forced by observed sea surface temperature) are documented here, as well as the major improvements with respect to the previous versions, which mainly come form the parametrization of tropical convection. A methodology is proposed to help analyse the sensitivity of the tropical Hadley–Walker circulation to the parametrization of cumulus convection and clouds. The tropical circulation is characterized using scalar potentials associated with the horizontal wind and horizontal transport of geopotential (the Laplacian of which is proportional to the total vertical momentum in the atmospheric column). The effect of parametrized physics is analysed in a regime sorted framework using the vertical velocity at 500 hPa as a proxy for large scale vertical motion. Compared to Tiedtke’s convection scheme, used in previous versions, the Emanuel’s scheme improves the representation of the Hadley–Walker circulation, with a relatively stronger and deeper large scale vertical ascent over tropical continents, and suppresses the marked patterns of concentrated rainfall over oceans. Thanks to the regime sorted analyses, these differences are attributed to intrinsic differences in the vertical distribution of convective heating, and to the lack of self-inhibition by precipitating downdraughts in Tiedtke’s parametrization. Both the convection and cloud schemes are shown to control the relative importance of large scale convection over land and ocean, an important point for the behaviour of the coupled model.

Appendix 1: about vertically integrated velocity potential

The purpose of this appendix is: (1) to express the vertical momentum of atmospheric columns as a z-weighted integral of the Laplacian of the velocity potential; (2) to show that it is also approximately the Laplacian of a z-weighted integral of the velocity potential. Only monthly mean velocity fields are considered and the scalar velocity potential at each level is chosen so that it is zero at the poles.

1.1 Vertical momentum

The vertical momentum $\tilde{w}$ of atmospheric columns reads:

$$ \tilde{w} = \int\limits_{z_s}^{\infty} {\text{d}}z \rho w \simeq - \int\limits_{z_s}^{\infty} {\text{d}}z {\frac{\omega} {g}} $$

(7)

where w and ω (≃ −ρ g w for these monthly mean fields) are the vertical velocity expressed in z and pressure coordinates, respectively, and z _s is the altitude of the surface.

Vertical integration of the continuity equation (taking into account $\vec{\nabla} \vec{V} = \nabla^2 \varphi$) yields an expression of ω in terms of the velocity potential:

$$ \omega_{(z)} - \omega_s = \int\limits_{z_s}^z {\text{d}}z^{\prime} \nabla^2 \varphi_{(z^{\prime})} \rho_{(z^{\prime})}g $$

(8)

For monthly mean fields, the term ${\omega_s = {{\partial p_s}/{\partial t}}}$ is negligible. Then, substitution of (8) in (7) yields:

$$ \tilde{w} = - \int\limits_0^{\infty} {\text{d}}z \int\limits_{z_s}^z {\text{d}}z^{\prime} \rho_{(z^{\prime})} \nabla^2 \varphi_{(z^{\prime})} $$

Let Z ₀ be an altitude high enough so that $\omega_{(Z_0)} \simeq 0.$ Then the integration may be limited to the triangle (z<Z ₀, z′<z). Permuting the two integrations yields:

$$\begin{aligned} \tilde{w} &= - \int\limits_{z_s}^{Z_0} {\text{d}}z^{\prime} \int\limits_{z^{\prime}}^{Z_0} {\text{d}}z \rho_{(z^{\prime})} \nabla^2 \varphi_{(z^{\prime})} \\ &= - \int\limits_{z_s}^{Z_0} {\text{d}}z^{\prime} \rho_{(z^{\prime})} \nabla^2 \varphi_{(z^{\prime})} (Z_0 - z^{\prime})\\ \end{aligned} $$

(9)

The Z ₀ term drops out since the vertical integral of ∇² φ is close to zero:

$$ \tilde{w} = {\frac{1} {g}} \int\limits_{p_0}^{p_s} {\text{d}}p z_{(p)} \nabla^2 \varphi_{(p)} $$

(10)

1.2 Expressing the vertical momentum $\tilde{w}$ in terms of the potential of the geopotential transport

We wish to express $\tilde{w}$ as the Laplacian of some potential. In order to do this, one has to commute the Laplacian operator in formula (10) with the z term and with the integral operator.

We shall limit ourselves to the tropical band where the geopotential altitude has weak horizontal variations. With such an approximation, the Laplacian and the z term commute.

Now we want to commute the horizontal differentials with the vertical integration. Taking into account the fact that the velocity is zero at the surface (so that $\vec{\nabla} \varphi(p_s) = 0$), one may write:

$$\begin{aligned} {\frac{\partial} {\partial x}} \left(\int\limits_{p_0}^{p_s} {\text{d}}p z\varphi \right) &= \int\limits_{p_0}^{p_s} {\text{d}}p z {\frac{\partial \varphi} {\partial x}} + z_{(p_s)} \varphi_{(p_s)} {\frac{\partial p_s} {\partial x}}\\ {\frac{\partial^2} {\partial x^2}} \left(\int\limits_{p_0}^{p_s} {\text{d}}p z\varphi \right) & = \int\limits_{p_0}^{p_s} {\text{d}}p z {\frac{\partial^2 \varphi} {\partial x^2}} + z_{(p_s)} \varphi_{(p_s)} {\frac{\partial^2 p_s} {\partial x^2}} \end{aligned}$$

(11)

Adding the analog formula for the y derivative, one gets:

$$ \nabla^2 \left(\int\limits_{p_0}^{p_s} {\text{d}}p z\varphi \right) = \int\limits_{p_0}^{p_s} {\text{d}}p z \nabla^2 \varphi + z_{(p_s)} \varphi_{(p_s)} \nabla^2 p_s $$

(12)

Over oceans, the last rhs term is zero. Over continents, it is not necessarily zero, because of orography. However, with a spatial resolution of the order of 100 km, it stays several order of magnitude smaller than the first rhs term and we shall neglect it.

Then, one may write the vertical momentum $\tilde{w}$ as the Laplacian of a function $\tilde{\varphi}:$

$$ \left\{ {\begin{array}{*{20}c} \tilde{w} \simeq {\frac{1} {g}} \nabla^2 \tilde{\varphi} \\ \tilde{\varphi} = \int\limits_{p_0}^{p_s} {\text{d}}p z \varphi \\ \end{array}} \right.$$

(13)

Finally, using the same technique and the same approximations one may prove that $\tilde{\varphi}$ is close to the saclar potential of the horizontal transport $\vec{G}$ of geopotential:

$$\begin{aligned} \vec{G} &= \int\limits_0^{\infty} {\text{d}}z \rho \vec{V} g z \\ & = \int\limits_0^{p_s} {\text{d}}p z \vec{V} \end{aligned}$$

(14)

As an illustration, Fig. 21 displays the potential $\tilde{\varphi}$ of the annual mean geopotential transport and the mean vertical velocity. The similarity of $\overline{\omega}$ and ω₅₀₀ is obvious. However, $\overline{\omega}$ is smoother than ω₅₀₀ and might be a better indicator of dynamic regimes. Finally, the lowest pannel illustrates the strong link between large-scale ascent and precipitation.

Fig. 21

Potential $\tilde{\varphi}$ of the annual mean of the horizontal transport of geopotential (upper pannel) and mean vertical velocity $\overline{\omega} = {\frac{1}{Z_0-z_s}} \int_{z_s}^{Z_0} {\text{d}}z \omega = {{-1}\over {Z_0-z_s}} {\nabla^2 \tilde{\varphi}}$ (with Z ₀ − z _s =15 km) (second pannel) for one of the AMIP simulations; to be compared with ω₅₀₀ and annual precipitation (two lower pannels)