# Perspectives for Very High-Resolution Climate Simulations with Nested Models: Illustration of Potential in Simulating St. Lawrence River Valley Channelling Winds with the Fifth-Generation Canadian Regional Climate Model

^{*}

## Abstract

**:**

## 1. Introduction

**Figure 1.**Geographical details of the study domain. The St. Lawrence River Valley (SLRV), Lake Champlain Valley (LCV), Lake Ontario and the Laurentian, Appalachian, Adirondack, Green and White mountains are indicated. The red and blue dots indicate the locations of Montréal and Québec City, respectively. Topographic height (meters) is shown by grey tones.

## 2. Experimental Design

#### 2.1. The Model Description

^{6}lateral diffusion.

#### 2.2. The Cascade Description

**Figure 2.**The computational domains used for the fifth-generation Canadian Regional Climate Model (CRCM5) cascade: d81 (0.81°; black), d27 (blue), d9 (red), d3 (green) and d1 (magenta) simulations. The domains are square and centered on Montréal (Québec, Canada). The grey tones represent the topography definition in each domain (see the legend of Figure 1).

## 3. Results

#### 3.1. Kinetic Energy Spectra

^{−3}power law, with k the horizontal wavenumber (e.g., [67,68]). Several studies indicate a transition to a spectral slope of k

^{−5/3}at the mesoscale (e.g., [69,70]); the theories explaining this part of the spectrum are not well understood, however, nor at what scale the transition between the two spectral slopes occurs.

_{U}

^{2}(k) and σ

_{V}

^{2}(k), are computed on pressure levels with the DCT as a function of the two-dimensional wavenumber k and then combined to form the KE spectrum ${\mathrm{\sigma}}_{\mathrm{K}\mathrm{E}}^{2}\left(\mathrm{k}\right)=\frac{{\mathrm{\sigma}}_{\mathrm{U}}^{2}(\mathrm{k})+{\mathrm{\sigma}}_{\mathrm{V}}^{2}(\mathrm{k})}{2}$, which can then be vertically integrated over some atmospheric depth and temporally averaged over some selected period. KE spectra of the various CRCM5 simulations have been computed and will be analyzed to study the spin-up process and the effective resolution of each simulation in the cascade.

#### 3.1.1. Spin-Up Time

^{2}), vertically integrated from 700 to 200 hPa; the upper abscissa indicates the wavelengths λ (in km) and the lower abscissa the wavenumbers k = 2π/λ (in radians km

^{−1}). With the DCT, the shortest wavelength or Nyquist length scale is twice the grid spacing (2 × ∆x), and the longest wavelength is twice the domain size (i.e., the number of grid points multiplied by the grid spacing, 2 × 50 × ∆x). The different line colors correspond to different times in the simulations.

**Figure 3.**Kinetic energy (KE) spectra at various times after the launching of the (

**a**) d81, (

**b**) d27, (

**c**) d9, (

**d**) d3 and (

**e**) d1 simulations. The different line colors correspond to different times in the simulation, as identified by the respective panels’ legends. The ordinates are the KE (J/m

^{2}) vertically averaged from 700 to 200 hPa; the upper abscissas represent the wavelengths (km) and the lower abscissas the wavenumbers (rad/km). All axes are shown on logarithmic scales.

#### 3.1.2. Effective Resolution

_{3 × n}coarser mesh simulation can be compared to the dashed line of the d

_{n}finer mesh simulation, with n ∈ {1, 3, 9, 27}, as these are computed over the same physical region, to define the effective resolution of a simulation using the nearest finer resolution spectrum as the reference.

^{−1}for d81, d27, d9 and d3 simulations, respectively. This gives effective resolution wavelengths (λ

_{eff}) of roughly 571, 190, 63 and 21 km, which correspond to roughly seven-times the grid spacing, i.e., 3.5 × (2 × ∆x). This finding is consistent with the numerical results obtained by Skamarock [72] for grid meshes of 22, 10 and 4 km and with numerical analysis theory that estimates the effective resolution of low-order finite-difference schemes to lie between 6 × ∆x and 10 × ∆x (e.g., [72]). Figure 4 also shows a shift in the spectral slopes from k

^{−3}at large scales to k

^{−5/3}at small scales in the d1 simulation. The transition occurs near k = 0.31 rad∙km

^{−1}, corresponding approximately to λ = 20 km. This value however needs to be taken with reservations, as it occurs close to the effective resolution of 7 km.

#### 3.2. St. Lawrence River Valley Features

**Figure 4.**Kinetic energy (KE) spectra for the d81 (black), d27 (blue), d9 (red), d3 (green) and d1 (pink) simulations. The abscissa represents the wavenumbers (rad/km), and the ordinate is the KE (J/m

^{2}), vertically averaged from 700 to 200 hPa and temporally averaged over the period from 0000 UTC 13 February to 0000 UTC 1 March 2002. The solid lines correspond to domains covering the inner 45 × 45 grid points, and the dashed lines correspond to the same spatial coverage in the next higher closest resolution (135 × 135 grid points). The arrows show the estimated effective resolution for each simulation. The grey and orange lines are the k

^{−3}and k

^{−5/3}characteristic spectral slopes, respectively.

**Figure 5.**Wind rose diagrams of the low-level horizontal wind for the (

**a**) d81, (

**b**) d27, (

**c**) d9 and (

**d**) d3 simulations, averaged over the period from 0000 UTC 4 February to 0000 UTC 7 March 2002, at the grid point closest to Québec City (blue dot in Figure 1); (

**e**) Surface wind roses diagrams of the Jean-Lesage Airport surface (© [76]) station averaged over the same period. The colors represent the wind intensities (m/s) and the circles the frequencies (%) for each intensity/direction pair.

**Figure 6.**(

**a**–

**d**) Same as Figure 5, but at the grid point closest to Montréal (red dot in Figure 1) and averaged over the period from 0000 UTC 13 February to 0000 UTC 1 March 2002. (

**e**) Same Figure 5, but for the Pierre-Elliott-Trudeau Airport surface station (© [76]) from the Environment Canada archives.

**Figure 7.**Typical situation for northeast winter wind channelling in the SLRV. Speed (m/s) and directions (background colors and black arrows, respectively) of the low-level wind and the mean sea level pressure (black dashed lines) for (

**a**) d81, (

**b**) d27, (

**c**) d9, (

**d**) d3 and (

**e**) d1 simulations, at 1200 UTC 26 February 2002; (

**f**) the corresponding analysed conditions (© [79]). The red square in (f) represents the region cover by the simulations’ results.

**Figure 8.**(

**a**) Vertical temperature profiles of the d81 (black line), d27 (blue line), d9 (red) and d3 (green line) simulations, for the grid point closest to Québec City (see the blue dot in Figure 1) at 1200 UTC 26 February 2002; (

**b**) corresponding observed vertical profile at the closest radio sounding station, Maniwaki, located some 500 km to the west of Québec City (© [81]).

## 4. Application to a Regional Climate Modeling Approach

_{GCM}(the cost) can be expressed as:

_{GCM}= K × N

_{x}× N

_{y}× N

_{z}× N

_{t}

_{x}and N

_{y}are the numbers of the grid points along the horizontal x- and y-axes, respectively, N

_{z}is the number of levels in the vertical and N

_{t}is the number of time steps required to complete the simulation. K is a complex function of the hardware, software and complexity of the model. The numbers of grid points N

_{x}and N

_{y}are directly linked to the horizontal grid spacing (Δx and Δy) given that the domain (L

_{x}and L

_{y}) is global. N

_{z}is related to the average vertical grid spacing (Δz) and to the height of the computational domain (L

_{z}), and finally, N

_{t}is connected to the time step (Δt) and the length of the simulation (L

_{t}); hence, Δ

_{i}= L

_{i}/N

_{i}for i ∈ {x,y,z,t}. Further assuming uniform vertical and horizontal grid spacings and Δx = Δy for simplicity, the cost of a GCM can be then expressed as:

_{FL}= U Δt/Δx be less than unity. In explicit Eulerian schemes, U is the sum of the maximum wind speed and the phase speed of the fastest waves, while in semi-implicit Eulerian schemes, U is the maximum wind speed. Semi-implicit semi-Lagrangian schemes allow relaxing this condition, and in practice, C

_{FL}≤ 5 is often used with acceptable accuracy [83]. In the following, we will consider for generality that the time step is Δt = C

_{FL}Δx/U, where C

_{FL}is an order unity quantity. With these constraints on Δz and Δt, the cost equation can be written as:

_{FL}are not under the direct control of the user. For a GCM, the horizontal domain (L

_{x}, L

_{y}) covers the entire globe, L

_{z}is heavily constrained to cover the troposphere and at least the lower stratosphere and the length of the simulation must be of the order of several decades to achieve equilibrium and statistical significance of the results. Hence, the well-known rule ensues: the cost of a GCM is inversely proportional to the fourth power of the horizontal grid spacing. As mentioned before, the condition on Δz is not always respected, and when N

_{z}and Δz are fixed arbitrarily, then the cost becomes an inverse function of the third power of the grid spacing:

_{x}× L

_{y}) is under the user’s control. A reduction of the domain size affords an important reduction of cost, which is what has made nested models increasingly popular tools over the last two decades. If one accepts using the same number of horizontal grid points (N

_{x}, N

_{y}) independently of the mesh size, hence reducing the domain size as the resolution increases, then the cost of a LAM simulation becomes inversely proportional to the first power of the grid spacing:

_{t}and Δx. The first rather than the third power dependency on grid spacing is what makes high-resolution nested models very competitive, as far as computational cost is concerned, compared to global models. Nevertheless, achieving convection-resolving simulations on a climate time scale still represents a humongous computational burden that is out of reach for most climate research groups, due to the linear inverse dependency on Δx.

_{t}/Δx constant, the cost would be constant, independent of mesh size:

_{FL}, N

_{z}, N

_{y}, N

_{x}and the ratio L

_{t}/Δx are kept constant, independent of mesh size; this last condition becomes very important for regional climate application, as discussed below.

^{η‑1}would be afforded for η times the cost of a single low-resolution simulation. The factor η represents a very benign increase of cost as a function of increased resolution compared to Equations (4)–(6).

_{t}/Δx) is perhaps the most severe, as statistical significance is of course an issue when the length of simulations is concerned. Nevertheless, [23] showed the utility of event-based downscaling of the ten most extreme 24-h precipitation events that occurred in 30-year periods, and this has inspired us the following thought experiment. Consider, for example, 30-year period as the canonical minimal period for studying general circulation in the atmosphere. Let us consider making a cascade of simulations ranging from a mesh size of 50-km down to 1 km. Maintaining a constant ratio of L

_{t}/Δx would imply that the 1-km simulation period would be 219 days long. If one were to accept to only activate the 1-km simulation for episodes of events of interests (such as thunderstorm, freezing rain events or wind storms) of one-day period at a time, with a few hours of spin-up each, this means that some 150 such episodes could be simulated with the 1-km mesh model. Given that these episodes could be chosen to be associated with a diversity of synoptic conditions spanning the 30 years, they would be independent of one another, and 150 samples would appear a very reasonable number to obtain statistical significance of the selected high-impact weather event of interest. The challenge then would be the identification of episodes of interest; these would have to be determined from the coarser mesh simulation, based on large-scale circulation patterns, convective available potential energy or the pressure gradient along the valleys, as used, for example, in the studies of [23,31,78,88].

## 5. Summaries and Conclusions

- The high-resolution simulations displayed marked improvements compared to coarser ones, because the mountain ranges in the vicinity of the SLRV are simply not resolved in coarser resolution simulations. Compared to coarser resolution (d81), the d3 and d1 topographies’ definitions of the CRCM5 were enhanced, and some of the highest mountains’ peak heights were doubled in the finer resolution.
- Kinetic energy spectra showed that small scales are better resolved with the finer resolution simulations and that the effective resolution wavelength is about seven-times the grid spacing of each simulation in the cascade.
- The high-resolution simulations succeeded in reproducing the known propensity of low-level winds to blow along the SLRV, despite the modest height of the bordering Laurentian and Appalachian mountain ranges. These valley winds were simply not resolved in coarser resolution simulations. For example, the wind direction shifted by 180° at the same grid point depending on the resolution.
- The vertical temperature structure is also impacted by the model horizontal resolution. For example, a simulation with a mesh of 81 km would lead to rain at the surface, whereas the 3‑km one would be associated with freezing rain. For instance, no refreezing layer and temperature inversion are found at lower levels for the simulation with grid spacing of 81 km. Furthermore, the depth and temperature of the melting layer varies significantly across model resolutions, which is directly linked to the type of precipitation.
- A pragmatic theoretical cost argument has been developed, suggesting a climatological framework to use the cascade method for studying specific high-impact weather of interest using very high-resolution regional climate modeling.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Cholette, M.; Laprise, R.; Thériault, J.M.
Perspectives for Very High-Resolution Climate Simulations with Nested Models: Illustration of Potential in Simulating St. Lawrence River Valley Channelling Winds with the Fifth-Generation Canadian Regional Climate Model. *Climate* **2015**, *3*, 283-307.
https://doi.org/10.3390/cli3020283

**AMA Style**

Cholette M, Laprise R, Thériault JM.
Perspectives for Very High-Resolution Climate Simulations with Nested Models: Illustration of Potential in Simulating St. Lawrence River Valley Channelling Winds with the Fifth-Generation Canadian Regional Climate Model. *Climate*. 2015; 3(2):283-307.
https://doi.org/10.3390/cli3020283

**Chicago/Turabian Style**

Cholette, Mélissa, René Laprise, and Julie Mireille Thériault.
2015. "Perspectives for Very High-Resolution Climate Simulations with Nested Models: Illustration of Potential in Simulating St. Lawrence River Valley Channelling Winds with the Fifth-Generation Canadian Regional Climate Model" *Climate* 3, no. 2: 283-307.
https://doi.org/10.3390/cli3020283