## 2 Materials and Methods
### Initial and Boundary Conditions  
Initial conditions (\(t=0\)) for air density: \(\rho(x,y,z,0)=\rho_{0}(x,y,z)\). For the city of Almaty, you can use the barometric formula, taking into account the altitude above sea level:  
\[\rho_{0}(x,y,z)=\rho_{sea}exp\left(-\frac{g\Big{(}z+h_{Almaty}\Big{)}}{RT_{0}} \right),\]  
where \(h_{Almaty}\approx 800\) m is the average height of Almaty above sea level. Velocity components are:  
\[u(x,y,z,0)=u_{0}(x,y,z),\]  
\[v(x,y,z,0)=v_{0}(x,y,z),\]  
\[w(x,y,z,0)=w_{0}(x,y,z).\]  
Temperature is \(T(x,y,z,0)=T_{0}(x,y,z)\). Almaty is characterized by a temperature gradient by height, as well as a temperature inversion in winter:  
\[T_{0}(x,y,z)=T_{surface}-\gamma z,\]  
where \(\gamma\) is the vertical temperature gradient. Pressure is \(P(x,y,z,0)=P_{0}(x,y,z)\). The initial pressure distribution can be specified in the hydrostatic approximation:  
\[P_{0}(x,y,z)=P_{surface}exp\bigg{(}-\frac{g(z)}{RT_{0}(z)}\bigg{)}.\]  
Concentration of pollutants:  
\[C(x,y,z,0)=C_{0}(x,y,z).\]  
The calculation domain is a parallelepiped \(\Omega=[0,L_{x}]\times[0,L_{y}]\times[0,L_{z}]\), where \(L_{x}\), \(L_{y}\), \(L_{z}\) are the dimensions of the domain in the corresponding directions. Boundary conditions at the lower boundary (ground, \(z=0\)):  
For the velocity, the no-slip condition \(u(x,y,0,t)=0\), \(v(x,y,0,t)=0\), \(w(x,y,0,t)=0\) are used. For the temperature of the earth, the surface temperature will be \(T(x,y,0,t)=T_{surface}(x,y,t)\). For the concentration at \(z=0\), there will be a given flow of pollutants \(-D\frac{\partial C}{\partial z}(x,y,0,t)=q_{\mathbb{C}}(x,y,t)\), where \(q_{\mathbb{C}}\) is the intensity of the emission of pollutants from the surface. For the velocity at the upper boundary (\(z=L_{z}\)) the free exit condition is set  
\[\frac{\partial u}{\partial z}(x,y,L_{z},t)=0,\ \frac{\partial v}{\partial z}(x,y, L_{z},t)=0,\ \frac{\partial w}{\partial z}(x,y,L_{z},t)=0.\]  
For the temperature at the upper boundary (\(z=L_{z}\)) the temperature gradient is specified:  
\[\frac{\partial T}{\partial z}(x,y,L_{z},t)=-\gamma_{atm},\]  
where \(\gamma_{atm}\) is the adiabatic temperature gradient in the atmosphere. For the concentration at the upper boundary (\(z=L_{z}\)), the free exit condition is set:  
\[\frac{\partial\mathbb{C}}{\partial z}(x,y,L_{z},t)=0.\]  
At the lateral boundaries, periodic boundary conditions or flow entry/exit conditions, depending on the wind direction, can be used. At the inlet boundary (for \(u>0\) at \(x=0\)):  
\[u(x,y,z,0)=u_{0}(x,y,z),\]0  
\[u(x,y,z,0)=u_{0}(x,y,z),\]1  
At the output boundary (for \(u>0\) at \(x=L_{x}\)):  
\[u(x,y,z,0)=u_{0}(x,y,z),\]2  
\[u(x,y,z,0)=u_{0}(x,y,z),\]3  
The values of the initial and boundary conditions--such as air temperature at the surface, pollutant concentration, wind velocity, pressure, and vertical temperature gradient--were determined based on the averaged historical data provided by the Kazakh Hydrometeorological Service (Kazhydromet) for the period 2015-2024. The vertical temperature gradient (\(\gamma\)) was set according to observed inversion profiles in winter and adiabatic lapse rates during summer months, as derived from balloon sounding data at the Almaty meteorological station. The initial surface temperature of T\({}_{0}\) = 17 \({}^{\circ}\mathbb{C}\) corresponds to the seasonal mean for early spring months, based on long-term climatic data. The pollutant concentration at ground level (C\({}_{0}\) = 0.8 g/m\({}^{3}\)) reflects average daily concentrations of PM\({}_{2.5}\) and PM\({}_{10}\) during episodes of high pollution recorded in central Almaty, as reported by Kazhydromet bulletins. Wind speed values at inlet boundaries were calibrated using monthly average wind roses and adjusted for altitude and urban roughness effects via barometric and logarithmic vertical wind profiles. These sources ensure that the model is grounded in real-world measurements and remains consistent with climatological and environmental norms for the city.