Diagnostic budget study of the internal variability in ensemble simulations of the Canadian RCM

Abstract

Due to the chaotic and nonlinear nature of the atmospheric dynamics, it is known that small differences in the initial conditions (IC) of models can grow and affect the simulation evolution. In this study, we perform a quantitative diagnostic budget calculation of the various diabatic and dynamical contributions to the time evolution and spatial distribution of internal variability (IV) in simulations with the nested Canadian Regional Climate Model. We establish prognostic budget equations of the IV for the potential temperature and the relative vorticity fields. For both of these variables, the IV equations present similar terms, notably terms relating to the transport of IV by ensemble-mean flow and to the covariance of fluctuations acting on the gradient of the ensemble-mean state. We show the skill of these equations to diagnose the IV that took place in an ensemble of 20 3-month (summer season) simulations that differed only in their IC. Our study suggests that the dominant terms responsible for the large increase of IV are either the covariance term involving the potential temperature fluctuations and diabatic heating fluctuations, or the covariance of inter-member fluctuations acting upon ensemble-mean gradients. Our results also show that, on average, the third-order terms are negligible, but they can become important when the IV is large.

Appendices

Appendix 1—Reynolds averaging and rules of averaging

Considering two variables f and g that are split into ensemble-mean and deviation parts:

$$ \begin{gathered} f = \left\langle f \right\rangle + f^{\prime} \hfill \\ g = \left\langle g \right\rangle + g^{\prime} \hfill \\ \end{gathered} $$

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Reynolds decomposition of the product gives:

$$ fg = \left\langle f \right\rangle \left\langle g \right\rangle + f^{\prime}\left\langle g \right\rangle + \left\langle f \right\rangle g^{\prime} + f^{\prime}g^{\prime} $$

(27)

so that,

$$ \left\langle {fg} \right\rangle = \left\langle f \right\rangle \left\langle g \right\rangle + \left\langle {f^{\prime}g^{\prime}} \right\rangle $$

(28)

and so,

$$ fg - \left\langle {fg} \right\rangle = f^{\prime}\left\langle g \right\rangle + \left\langle f \right\rangle g^{\prime} + f^{\prime}g^{\prime} - \left\langle {f^{\prime}g^{\prime}} \right\rangle $$

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The following results from 26:

$$ \begin{gathered} \left\langle {f^{\prime}} \right\rangle = 0; \hfill \\ \left\langle {\left\langle f \right\rangle g} \right\rangle = \left\langle f \right\rangle \left\langle g \right\rangle ; \hfill \\ \left\langle {f + g} \right\rangle = \left\langle f \right\rangle + \left\langle g \right\rangle \hfill \\ \end{gathered} $$

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If α is a constant, we also get;

$$ \left\langle {\alpha f} \right\rangle = \alpha \left\langle f \right\rangle $$

(31)

From the definition of partial derivatives:

$$ \left\langle {{\frac{\partial f}{\partial \xi }}} \right\rangle = {\frac{\partial \left\langle f \right\rangle }{\partial \xi }} $$

(32)

where ξ represent space or time variable.

Appendix 2—Relative vorticity equation ζ for the polar stereographic and the vertical pressure coordinates

Consider a system of the polar stereographic projection(X, Y) and the vertical pressure coordinates, the Euler equations, with the meteorological “traditional” approximation, take the following form:

$$ {\frac{\partial U}{\partial t}} + \overrightarrow {V} \cdot \overrightarrow {\nabla } U + \omega {\frac{\partial U}{\partial p}} = fV - K{\frac{\partial S}{\partial X}} - {\frac{\partial \phi }{\partial X}} + F_{X} - c\nabla^{4} U $$

(33)

$$ {\frac{\partial V}{\partial t}} + \overrightarrow {V} \cdot \overrightarrow {\nabla } V + \omega {\frac{\partial V}{\partial p}} = - fU - K{\frac{\partial S}{\partial Y}} - {\frac{\partial \phi }{\partial Y}} + F_{Y} - c\nabla^{4} V $$

(34)

where the different variables of these equations are defined as:

$$ \left( \begin{gathered} dX \hfill \\ dY \hfill \\ \end{gathered} \right) = m \times \left[ \begin{gathered} - \sin \lambda {\text{ - cos}}\lambda \hfill \\ { \cos }\lambda {\text{ - sin}}\lambda \hfill \\ \end{gathered} \right]\left( \begin{gathered} dx \hfill \\ dy \hfill \\ \end{gathered} \right) $$

with λ the longitude and (x, y) the local horizontal Cartesian coordinate system.

$K = {{\left( {U^{2} + V^{2} } \right)} \mathord{\left/ {\vphantom {{\left( {U^{2} + V^{2} } \right)} 2}} \right. \kern-\nulldelimiterspace} 2}$ and S = m² with m = (1 + sin ψ₀)/(1 + sin ψ) the map scale factor, where ψ is the latitude and ψ₀ the latitude at which the plane of projection intersects the sphere.

(F _X , F _Y ) correspond to components of sources or sinks of momentum. These two components include surface friction, vertical diffusion and gravity wave drag. The last term of equations 33 and 34 takes into account the horizontal diffusion to parameterise the cascade of information to unresolved scale, where the constant of diffusion c for the 45-km version of the CRCM is c = 1.85.10¹³ m⁴ s⁻¹ (Laprise et al. 1998).

f = 2Ω sinψ is the Coriolis parameter.

By adding and subtracting to equations 33 and 34 these two expressions: $SV\left[ {{{\partial V} \mathord{\left/ {\vphantom {{\partial V} {\partial X}}} \right. \kern-\nulldelimiterspace} {\partial X}}} \right]$ and $SU\left[ {{{\partial U} \mathord{\left/ {\vphantom {{\partial U} {\partial Y}}} \right. \kern-\nulldelimiterspace} {\partial Y}}} \right]$ respectively, we obtain:

$$ {\frac{\partial U}{\partial t}} + \overrightarrow {V} \cdot \overrightarrow {\nabla } U + \omega {\frac{\partial U}{\partial p}} = \left( {\zeta + f} \right)V - {\frac{{\partial \left( {KS} \right)}}{\partial X}} - {\frac{\partial \phi }{\partial X}} + F_{X} - c\nabla^{4} U $$

(35)

$$ {\frac{\partial V}{\partial t}} + \overrightarrow {V} \cdot \overrightarrow {\nabla } V + \omega {\frac{\partial V}{\partial p}} = - \left( {\zeta + f} \right)U - {\frac{{\partial \left( {KS} \right)}}{\partial Y}} - {\frac{\partial \phi }{\partial Y}} + F_{Y} - c\nabla^{4} V $$

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where $\zeta = S\left[ {{\frac{\partial V}{\partial X}} - {\frac{\partial U}{\partial Y}}} \right]$ is the relative vorticity expression.

Applying $S \times \left[ {{\frac{{\partial \left( {B - 2} \right)}}{\partial X}} - {\frac{{\partial \left( {B - 1} \right)}}{\partial Y}}} \right]$, we obtain the relative vorticity equation:

$$ {\frac{\partial \zeta }{\partial t}} + \overrightarrow {\nabla } \cdot \left( {\zeta \overrightarrow {V} } \right) + \omega {\frac{\partial \zeta }{\partial p}} = - \overrightarrow {\nabla } \cdot \left( {f\overrightarrow {V} } \right) + S\left[ {{\frac{\partial \omega }{\partial Y}}{\frac{\partial U}{\partial p}} - {\frac{\partial \omega }{\partial X}}{\frac{\partial V}{\partial p}}} \right] + S\left[ {{\frac{{\partial F_{Y} }}{\partial X}} - {\frac{{\partial F_{X} }}{\partial Y}}} \right] - c\nabla^{4} \zeta $$

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Appendix 3—Spatial and temporal finite differencing

To derive second-order finite-difference equations, the following sum and difference operations are defined (e.g. Haltiner and Williams 1980; Durran 1999):

$$ \delta_{t} F\left( {i,j,k,t} \right) \equiv {\frac{{F\left( {i,j,k,t + 1} \right) - F\left( {i,j,k,t - 1} \right)}}{2\Updelta t}} $$

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$$ \begin{gathered} \overline{{F\left( {i,j,k,t} \right)}}^{X} \equiv {\frac{{F\left( {i + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2},j,k,t} \right) + F\left( {i - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2},j,k,t} \right)}}{2}} \hfill \\ \overline{{F\left( {i,j,k,t} \right)}}^{Y} \equiv {\frac{{F\left( {i,j + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2},k,t} \right) + F\left( {i,j - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2},k,t} \right)}}{2}} \hfill \\ \end{gathered} $$

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$$ \begin{gathered} \delta_{X} F\left( {i,j,k,t} \right) \equiv {\frac{{F\left( {i + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2},j,k,t} \right) - F\left( {i - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2},j,k,t} \right)}}{\Updelta X}} \hfill \\ \delta_{Y} F\left( {i,j,k,t} \right) \equiv {\frac{{F\left( {i,j + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2},k,t} \right) - F\left( {i,j - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2},k,t} \right)}}{\Updelta Y}} \hfill \\ \end{gathered} $$

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$$ \begin{gathered} \delta_{X} \overline{{F\left( {i,j,k,t} \right)}}^{X} \equiv {\frac{{F\left( {i + 1,j,k,t} \right) - F\left( {i - 1,j,k,t} \right)}}{2\Updelta X}} \hfill \\ \delta_{Y} \overline{{F\left( {i,j,k,t} \right)}}^{Y} \equiv {\frac{{F\left( {i,j + 1,k,t} \right) - F\left( {i,j - 1,k,t} \right)}}{2\Updelta Y}} \hfill \\ \end{gathered} $$

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$$ \delta_{p} F\left( {i,j,k,t} \right) \equiv {\frac{{F\left( {i,j,k + 1,t} \right) - F\left( {i,j,k - 1,t} \right)}}{{p\left( {k + 1} \right) - p\left( {k - 1} \right)}}} $$

(42)

$$ \nabla \cdot \overrightarrow {V} \approx S\left[ {\delta_{X} \overline{U}^{X} + \delta_{Y} \overline{V}^{Y} } \right]{\text{ where }}\overrightarrow {V} \equiv \left( {U,V} \right) $$

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Appendix 4—Different terms of the IV prognostic equation for the relative vorticity

$$ L_{\zeta } \left( {i,j,k,t} \right) = R_{\zeta } \left( {i,j,k,t} \right) $$

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where,

$$ \begin{gathered} L_{\zeta } \left( {i,j,k,t} \right) \approx \delta_{t} \sigma_{\zeta }^{2} \left( {i,j,k,t} \right) \hfill \\ \approx {\frac{{\sigma_{\zeta }^{2} \left( {i,j,k,t + 1} \right) - \sigma_{\zeta }^{2} \left( {i,j,k,t - 1} \right)}}{2\Updelta t}} \hfill \\ \end{gathered} $$

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and,

$$ R_{\zeta } \left( {i,j,k,t} \right) = \left[ {A_{h} + A_{v} + N_{h} + B_{h} + B_{v} + C + D + E_{h} + F_{v} + G + H} \right]\left( {i,j,k,t} \right) $$

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with

$$ \begin{gathered} A_{h} \approx - S\left[ {\delta_{X} \left( {\overline{\left\langle U \right\rangle }^{X} \overline{{\sigma_{\zeta }^{2} }}^{X} } \right) + \delta_{Y} \left( {\overline{\left\langle V \right\rangle }^{Y} \overline{{\sigma_{\zeta }^{2} }}^{Y} } \right)} \right]; \, A_{v} \approx - \delta_{p} \left( {\left\langle \omega \right\rangle \sigma_{\zeta }^{2} } \right) \hfill \\ N_{h} \approx - 2S\sigma_{t}^{2} \left[ {\delta_{X} \overline{\left\langle U \right\rangle }^{X} + \delta_{Y} \overline{\left\langle V \right\rangle }^{Y} } \right]; \hfill \\ B_{h} \approx - 2S\left[ {\left\langle {\zeta^{\prime}_{n} U^{\prime}_{n} } \right\rangle \delta_{X} \overline{\left\langle \theta \right\rangle }^{X} + \left\langle {\zeta^{\prime}_{n} V^{\prime}_{n} } \right\rangle \delta_{Y} \overline{\left\langle \zeta \right\rangle }^{Y} } \right] ; {\text{ B}}_{v} \approx - 2\left\langle {\zeta^{\prime}_{n} \omega^{\prime}_{n} } \right\rangle \delta_{p} \left\langle \zeta \right\rangle \hfill \\ C \approx - 2S\left\langle \zeta \right\rangle \left[ {\left\langle {\zeta^{\prime}_{n} \delta_{X} \overline{{U^{\prime}_{n} }}^{X} } \right\rangle + \left\langle {\zeta^{\prime}_{n} \delta_{Y} \overline{{V^{\prime}_{n} }}^{Y} } \right\rangle } \right] ;\hfill \\ D \approx - 2S\left[ {\left\langle {\zeta^{\prime}_{n} \delta_{X} \left( {\overline{f}^{X} \overline{{U^{\prime}_{n} }}^{X} } \right)} \right\rangle + \left\langle {\zeta^{\prime}_{n} \delta_{Y} \left( {\overline{f}^{Y} \overline{{V^{\prime}_{n} }}^{Y} } \right)} \right\rangle } \right] ;\hfill \\ {\text{E}}_{h} \approx 2S\left[ {\delta_{Y} \overline{\left\langle \omega \right\rangle }^{Y} \left\langle {\zeta^{\prime}_{n} \delta_{p} U^{\prime}_{n} } \right\rangle - \delta_{X} \overline{\left\langle \omega \right\rangle }^{X} \left\langle {\zeta^{\prime}_{n} \delta_{p} V^{\prime}_{n} } \right\rangle } \right] ;\hfill \\ F_{v} \approx 2S\left[ {\left\langle {\zeta^{\prime}_{n} \delta_{Y} \overline{{\omega^{\prime}_{n} }}^{Y} } \right\rangle \delta_{p} \left\langle U \right\rangle - \left\langle {\zeta^{\prime}_{n} \delta_{X} \overline{{\omega^{\prime}_{n} }}^{X} } \right\rangle \delta_{p} \left\langle V \right\rangle } \right] ;\hfill \\ G \approx 2\left[ {S\left( {\left\langle {\zeta^{\prime}_{n} \delta_{X} \overline{{F^{\prime}_{Yn} }}^{X} } \right\rangle - \left\langle {\zeta^{\prime}_{n} \delta_{Y} \overline{{F^{\prime}_{Xn} }}^{Y} } \right\rangle } \right) - c\left\langle {\zeta^{\prime}_{n} \nabla^{4} \zeta^{\prime}_{n} } \right\rangle } \right] ;\hfill \\ H \approx - 2S\left[ {\left\langle {\zeta^{\prime}_{n} \delta_{X} \left( {\overline{{\zeta^{\prime}_{n} }}^{X} \overline{{U^{\prime}_{n} }}^{X} } \right)} \right\rangle + \left\langle {\zeta^{\prime}_{n} \delta_{X} \left( {\overline{{\zeta^{\prime}_{n} }}^{X} \overline{{V^{\prime}_{n} }}^{X} } \right)} \right\rangle } \right] - \left\langle {\omega^{\prime}_{n} \delta_{p} \zeta_{n}^{\prime 2} } \right\rangle \hfill \\ + 2S\left[ {\left\langle {\zeta^{\prime}_{n} \delta_{Y} \overline{{\omega^{\prime}_{n} }}^{Y} \delta_{p} U^{\prime}_{n} } \right\rangle - \left\langle {\zeta^{\prime}_{n} \delta_{X} \overline{{\omega^{\prime}_{n} }}^{X} \delta_{p} V^{\prime}_{n} } \right\rangle } \right] \hfill \\ \end{gathered} $$

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Appendix 5—Assessment of the vertical motion: $\omega_{n} \, \left( { \equiv \frac{dp}{dt}} \right)$

By definition, the vertical velocity can be written in this form:

$$ w = \frac{dz}{dt} = \frac{1}{g}{\frac{d\phi }{dt}} $$

(48)

where $\phi = gz$ represents the geopotential height. Expanding the total derivative into its components gives

$$ w = \frac{1}{g}\left[ {\left. {{\frac{\partial \phi }{\partial t}}} \right|_{p} + \overrightarrow {{V_{h} }} \cdot \nabla_{p} \phi + \omega {\frac{\partial \phi }{\partial p}}} \right] $$

(49)

Using the hydrostatic equation, we can write: ${\frac{\partial \phi }{\partial p}} = - \frac{RT}{p}$, so

$$ \omega = \frac{p}{RT}\left[ {\left. {{\frac{\partial \phi }{\partial t}}} \right|_{p} + \vec{V}_{h} \cdot \nabla_{p} \phi - gw} \right] $$

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All variables in this last equation are archived as the CRCM output data. Thus, the vertical motion is assessed as follow:

$$ \omega \approx \frac{p}{RT}\left[ {\delta_{t} \phi + S\left( {U\delta_{X} \overline{\phi }^{X} + V\delta_{Y} \overline{\phi }^{Y} } \right) - gw} \right] $$

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Thus, the vertical motion for the member n is written as:

$$ \omega_{n} \approx {\frac{p}{{RT_{n} }}}\left[ {\delta_{t} \phi_{n} + S\left( {U_{n} \delta_{X} \overline{{\phi_{n} }}^{X} + V_{n} \delta_{Y} \overline{{\phi_{n} }}^{Y} } \right) - gw_{n} } \right] $$

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