Influence of temperature perturbation on moisture dynamics

We analyzed the perturbation effect of temperature on the moisture content in the atmosphere [], considering the saturation phenomenon. The air parcels of a saturated atmosphere are made of dry air, vapor, cloud, and rain water that flow on a rotating Earth, with a temperature perturbations induced at 9°1′48″ N, 38°44′24″E and 6.324 km above the Earth’s surface. The fundamental atmospheric parameter of this saturated moist air was treated as temperature-dependent, and the pressure force exerted on it by the precipitation and cloud water is not considered in the numerical computation. In this proposed research, the initial value of wind speed along zonal, meridional, and vertical directions varies with elevation, and the interaction of the atmosphere with the Earth’s surface is not neglected. The result of the investigation revealed that a single tropospheric disturbance of an atmospheric variable is a major factor in the spread of different long period waves patterns (350 hours) and this atmospheric oscillations play a crucial role in transferring energy, momentum, and mass across different parts of the atmosphere, impacting global climate systems.

α v , β v parameters defined by the formula (8) δ ik ,ǫsress tensor,a unit dyadic (the gradient of the position vector) * * saurated vapor pressure at equilibrium and perturbed state defined by formulas (7) and (11) p do , ρ do equilibrium pressure and density of dry air Legendre polynomial of n degree Q , l ¢ F latent heat relase and the friction dissipation of the saturated moist air that defined by the formulas (25) and (27) q j mixing ratio of the saturated moist amosphere r r ,

Introduction
The majority of the atmosphere is concentrated close to Earth's surface, within ten kilometers of the places where clouds gather and aircraft take off [2,3].Considering hydrological circulation in the Earth's atmosphere, it is best understood as a moist atmosphere [4], which contains several phases of water material and the interactions between these phases of water are essential to mesoscale moist circulation [5][6][7].One result of phase transitions in the atmosphere is the formation of clouds and precipitation [8].According to [9][10][11], clouds and precipitation are one of the main sources of ambiguity in weather prediction and climate modeling, and the incorporation of moisture and phase transitions in atmospheric flow models is still actively debated [12].
Visible clouds cover more than half of the planet [13] and it is referred to as the 'regulator' of radiation balance in the Earth-atmosphere system [2] and is crucial to the planetary energy and water cycles [14,15].Furthermore, clouds act as a shield against long wave radiation that is released by continents and oceans, absorbing it and re-emitting part of it back into space.This process keeps the planet's average temperature constant and produces the atmosphere's greenhouse effect [16][17][18][19].
Different-period waves interact with atmospheric disturbances brought about by the physical mechanisms behind the formation of cloud and precipitation [20][21][22] and according to experimental evidence, the response of the upper atmosphere occurs several hours after disturbances near the Earth's surface [23][24][25][26].As a result, clouds and precipitations have a significant modulating effect on weather and climate scales [27][28][29][30][31]. Without considering the interaction of the atmosphere with various surfaces, the numerical computation of the perturbation effect of temperature and moist air density on lower atmospheric dynamics was done [1].The findings showed that every variable responded to the dynamics of the atmosphere in a plane wave pattern, with specific heat capacities, resulting wind speed, and water vapor mixing ratio increasing over time at each latitude, while vertical wind speed, specific enthalpy, and pressure decreased over time at each latitude.
In our previous work [1], we treated the atmosphere as unsaturated moist air, meaning that it simply consists of the components of gas and vapor water.But the main sources of uncertainty in weather forecasting are clouds and precipitation [32][33][34] and their influence on the saturated atmosphere's large-scale dynamics was not taken into account by the model.This work aims to understand the effects of a single temperature change on the dynamics of a saturated atmosphere within the Ethiopian atmosphere.To comprehend the fundamental equations of saturated moist air, this work also takes into account the interaction of clouds, rain, and dry air with vapor.Those equations govern the dynamics of the model and they must be approximated by some suitable discretization [35].
The discretization were carried out with a finite difference method on an unstaggered gird cell for all atmospheric parameters and the result of the investigation revealed that a single local disturbance of an atmospheric variable is a major factor in the spread of different wave patterns and this atmospheric oscillations play a crucial role in transferring energy, momentum, and mass across different parts of the atmosphere, impacting global climate systems.The remainder of this paper is structured as follows: The next section presents the model.Here, we provide a detailed description of the primitive equations.Section 4 discusses the numerical methods.The results and discussion are presented in section 5.For this, we use data obtained from numerical computation using FDM on an unstaggered grid cell for all atmosphere parameters.Section 6 concludes this article.

The theoretical model
For the purposes of the current work, we consider the atmosphere to be a neutral, compressible, and saturated moist air (cannot be supersaturated) moving in the rotating Earth's frame.Due to a different activities such as a volcanic eruption, large scale wild fire of forest, large scale heat releasing activities in town that was discussed in our preceding works [1], there was uncertainty of a moist air temperature that was induced at a particular location of the atmosphere (see, section 4).
The atmosphere is a complex fluid system [36][37][38][39] and to simplify this complexity we consider only the interaction of air parcel with ground surface with neglecting any pressure force that exerted on the air parcel by liquid water.Additionally, we focus only on warm clouds, where water is present only in gaseous and liquid form, i.e. no ice and snow phases occur in this theoretical model.
The gas constant (R m ), specific heat capacities at a constant pressure (c p ), enthalpy (h m ), and entropy (s m ) of the saturated moist atmosphere vary with the mixing ratio of the air z t q , , , ( ) l f j (values of j are 'd' for dry air, 'v' for water vapor, 'c' for cloud water and 'r' for rain) as shown in below.
where, λ, f, z and t stand for longitude, latitude, elevation above the Earth surface, and the time taken.The term R d is the individual gas constant for dry air and q v , q c , q r denote the mixing ratio of vapor, cloud and rain water, respectively.Moreover, c pd , c pv ,c vd , c vv are the specific heat of dry air and water vapor at constant pressure and volume, respectively.But c l represent the specific heat of liquid water and their numerical value is given in table 1.
The actual state of the saturated moist air subjected to the perturbation of an atmosphere temperature can be written as the sum of equilibrium and perturbed state [40][41][42].
Temperature of the air at a different location have been presented by [43,44], and the equilibrium temperature T mo of the moist air is obtained based on latitudinal varying tropospheric temperature lapse rate [45]: Where a, b, and c are constants and their numerical value used in this calculation is given in appendix 1.On the other hand, if a saturation effects occur on a moist air, then the mixing ratio of the system at the equilibrium state is the summation of the mixing ratio a dry air, water vapor, cloud water and the rain: Here, the mixing ratio for each component of the air is given as follows q z q z q z q z r z q z q z q z r z q z q z q z q z , , Using the condition of saturation [4], we have the following relation for total mixing ratio of water content q To (λ, f, z) and the saturated molar mixing ratio r * (λ, f, z): To vo mo vo vo mo vo is the saturation vapor pressure at equilibrium state and given by with T mo (λ, f, z) as in equation (3) and the constants α v , β v are given by We use the following representative values for the universal gas constant for a vapor water R v , sea level temperature T sl , latent heat LH o and the saturation vapor pressure p o * at temperature 0 °C (table 1) Additionally, the equilibrium pressure p mo (λ, f, z) and density ρ mo (λ, f, z) that was mentioned in equation (2) obtained by relating the thermodynamic quantities for a moist atmosphere via the ideal gas law as follows:  with q vo (λ, f, z) as in equation ( 5) and q To (λ, f, z) as in (6) .The detail derivation of dry air pressure p do and its corresponding density ρ do is given in appendix (31) and (32).Inserting equation ( 5) and ( 6) into (9) yields the following relation for the equilibrium pressure Due to a disturbance of temperature on the atmosphere, there was a perturbation saturated vapor pressure which is given as And the perturbed pressure can be obtained by using equation (11) into (10).
The corresponding perturbed parts of the mixing ratio of the air is becomes Using equation (2) on the ideal relation, the total density of the moist atmosphere can be expressed as and T m (λ, f, z, t) as in equations (1) and (3), respectively.Whereas, the total pressure of the air p m (λ, f, z, t) is obtained by taking the sum of equations (10) and (12).Finally, the perturbed density of the system is calculated as follows: with ρ mo (λ, f, z) and ρ m (λ, f, z, t) as in equations ( 9) and (14), respectively.

Thermodynamic equation
When modelling atmospheric flows in general, the full compressible governing equations need to be considered.These a coupled set of differential equations were obtained based on the following principles

Conservation of mass of moist air
Without omitting the contribution of the gas component by diffusion in this section, the basic equations for continuity of dry air q d , water vapor mixing ratio q v , cloud water mixing ratio q c , and rain water mixing ratio q r are given by Here T m and p m were given earlier in equations (3) and (14).The operator .. D Dt ( ) is the advective derivative of the j-th component of a saturated moist air in spherical coordinates.Other physical quantities include S ev , S cd , S ac , and S cr which are respectively the rates of evaporation of rain water, the condensation of water vapor to cloud, the auto-conversion of cloud water into rainwater by the accumulation of microscopic droplets, and the collection of cloud water by falling rain.Their corresponding parametric formulas are given in (A.1) and (A.2).Furthermore, v rain  and  q r denote the terminal velocity of the precipitating component relative and diffusion flux of the air, see their expression in appendixs (A19) and (A20).
Based on a Sutherlands law [46][47][48], the saturated moist air has temperature dependent transport coefficient (k T and μ T ) and as mentioned in our preceding work [ here, T m the sum of the equilibrium (T mo ) and perturbed T m ( ) ¢ temperature of the moist, with as in (T mo ) and T m ( ) ¢ as in equation (3).

Conservation of momentum of moist air
As the components of liquid water interact with the dry air, the equations of motion for saturated moist air subject to Coriolis, centrifugal, gravitational, pressure, and viscous forces are written as here, v  is the velocity of moist air that varies with the geometrical height(z) of the atmosphere as explain in equation (28) .The first term, q v z r rain to right side to account the transports of momentum due to rain and v rain  is introduced in (A3).The term  W and r  represent the angular velocity and r  is the position from the center of the Earth surface to air parcels.In [49], it is justified that the Earth's acceleration owing to gravity, g  , changes with the atmosphere's geometric position.
where P 2 , P 3 , and P 4 are Legendre polynomials of degrees 2, 3, and 4, respectively, that can be produced in a gravity model using recurrence formulas [49].Moreover, the last right-hand side of equation ( 18) is the viscous force that is exerted on the saturated moist air, and in terms of the stress tensor (δ ik ), it can be expressed as For a Newtonian fluid, the stress tensor of rank two is provided in [50].
The transposition operation is represented by the superscript T and μ T as in equation (17).The frictional force f vis where the viscous forces acting on the air parcel as a result of temperature-dependent and independent dynamic viscosity are denoted by f  m and f o  , respectively.

Conservation of energy of air parcel
We describe the thermodynamic equation using the equation of moist internal energy found in [4]: According to [6,51,52], the specific internal energy e of the moist air is the additive function, which is defined as e z t q e q e q e qe where e c T z t e c T z t L e e cT z t where we have defined , which represents the latent heat at the temperature 0 K.The term c vd , c vv , and c l are the heat capacities at constant volume for dry air, vapor and liquid water, respectively and their corresponding value is given in table 1.On the left side of equation (23), rain-induced internal energy transport is represented by q e v m r r rain  r , while precipitation-induced heat release is represented by g q v m r rain   r on the right.
The third term (Q H ¢ ) on the right-hand side in equation (23) represents the convergence of energy fluxes, which include the frictional dissipation of moist air motion (Φ) and latent heat release Where R v is the universal gas constant of water vapor, β v as defined in equation (8), S ac and S ev are already given in appendix (A1).
The dissipation function resulting from the viscous force on the air parcel (Φ) can be expressed as the double dot product of the stress tensors (σ ik ) and ( v    ) [1].Thus, by employing a tensor identity, Φ can be expressed as: Here is the relation for Φ that we acquire by introducing equations (17) into (21) and then again the result into equation (26).
The last term q •    on right-hand side of equation (23) which indicate the conduction heat flux and its mathematical expression was given in appendix (A18).

Numerical analysis
We consider a saturated moist atmosphere with four components, dry air, water vapor, cloud water, and rain subjected to a local generated perturbation of temperature T m (λ, f, z, t) = 0.025T mo (λ, f, z) that was induced at 9 1 48 o ¢ N, 38 44 24   ¢  E and 6.324 km above the Earth's surface at an initial time t = 0.For this specific work, the equilibrium temperature T mo (λ, f, z) of the saturated moist air was obtained by solving equation (3) that varies with only the spatial variables f and z but don't change with longitude λ and time t.The overall atmospheric model simulation processes for this specific work are described by the partial differential equations ( 16), ( 18), (23),and the ideal gas relation (14).By numerically computing those equations with the use of the finite difference method (FDM) [1] on an unstragered grid cell, key atmospheric parameters that responded to the temperature perturbation were extracted at a specific longitude and elevation above the Earth's surface.The middle troposphere, which is situated below 7 km and within the latitude range of 9 o N to 15 o N and the longitude range of 35 o E to 45 o E, is the domain size employed in the numerical computation.
In order to adhere to the Courant-Friedrich-Lewy condition, the computational grid size along the longitude, latitude, altitude, and associated temporal time steps are as follows: Δλ = 0.05 o , Δf = 0.05 o , Δz = 20 meters, and Δt = 3 minutes, separately.As we described in section 3.2, all the forces that mentioned in Navier-Stokes equation (18) have their own effect on the dynamics of air parcels.Due to this, the wind speed profile over the laminar sub-layer that can be described as a trigonometric function of elevation [53] v U z tanh 10 28 Here, the coefficient U has three components and near to the Earth's surface the initial value of wind speed along the zonal, meridional, and radial directions are u = 10 ms −1 , v = 10 ms −1 , and w = 0.1 ms −1 , respectively.
The purpose and essential of performing this numerical computations is to produce numerical values for the atmospheric variables as a function of geometrical position (λ, f, z) and time (t), which will help to understand the initial equilibrium atmosphere's response to changes in temperature of saturated wet air.Python, the program required for data charting, was used to extract the pertinent data at each λ and to analyze the data that resulted.

Results
This section shows a graphical representation of the atmospheric variables for the purpose of discussing the effect of a local perturbed temperature.It is shown from the first two figures 1 and 2 that both horizontal (v r ) and vertical (w) wind speeds of the moist air are increasing with time but constant along each latitude f at the mentoring point.The reason for this rise in v h and w could be attributed to the rain ( ) that is provided by equation (18), as well as a mechanical energy gain at the expense of internal energy [1].
After perturbation, the temperature T m of a saturated moist air is calculated using equation (24) and the result of T m function of time and latitude at a specific point of λ and z is depicted in figure 3.This finding shows that T m is not a constant but rather sharply falls with t at each f.This decreasing of T m at each f is the collective effect of precipitation-induced heat and rain-induced internal energy in equation (23).As we can see from the above figures 1, 2, and 3, the wind speeds of the moist air have an inverse relationship with the temporal propagation of temperature.This relation shows how the dissipation function Φin equation (23) causes the transformation of energies from one form to another throughout the system.Figure 4 illustrates how the perturbation of temperature affects the entropy of the saturated moist air.From this figure, we have seen that there is a decreasing of entropy s m as function of time at each f.This result is the consequence of decreasing T m with time (see figure 3).We noted that with a decrease in temperature, randomness (entropy) decreases because the motion of the particles decreases and velocity decreases, so they have less entropy at lower temperature profiles, which is clearly observed in figure 4.   To determine the effect of perturbation on the pressure of the moist air, we substitute equations (10) and ( 12) into (2).and the result of the numerical computation is shown in figure 5. From this result, we have seen that the propagation of pressure with respect to time is similar to the propagation of entropy and temperature in air parcels.Physically, as temperature decreases, the air parcels compress, and their molecules move slowly and possess less energy.As a result of this, less pressure is exerted on the saturated moist air.Equation (1) can be quantitatively solved to predict the perturbation's reaction to a gas constant R m and a specific heat capacity c pm at each time step.The corresponding plots of the data that were generated numerically are shown in figures 6 and 7, respectively.We find that the R m wave pattern is comparable to T m , which has a constant value at each f and diminishes with time.This outcome represents that the R m is a highly dependent variable on the T m of the saturated moist air (the mixing ratio of the moist air is a function of temperature T m ).Additionally, the c pm of the air parcels also behaves like a wave that rises with time at a mentoring point after perturbation.
Following a numerical calculation of the continuity equation ( 16), we obtain the densities for each moist air component (ρ d , ρ v , ρ c , ρ r ), and using equation (14), the net density of the system as a function of time and latitude at a mentoring point is displayed in figures 8. The findings that represented in figures 3, 5, 6, and 8 demonstrate that along the time direction higher temperature, pressure, and gas constant lead to lower density, and concurrent decreases in these parameters result in higher density.

Conclusions
This study revealed the perturbation effect of temperature on a neutral, compressible, and moist saturated air that is moving on the rotating Earth's surface.The components of a saturated atmosphere that are examined here consist of water vapor, cloud water, rainwater, and dry air.The pressure force exerted by clouds and rainwater on the saturated moist air has no significant effect, and it is ignored in numerical computation.In addition to the gas constant R m , the system's specific heat capacity c pm , enthalpy h m , and entropy s m are temperature-dependent parameters that change in response to the mixing ratio q j (λ, f, z, t).
The numerical method and condition of stability used for computing the effect of perturbation are similar to our previous paper [1].In this work, the initial value of wind speed varies with the elevation (z) of the  atmosphere, and the interaction of the atmosphere with the ground surface is not neglected.So, the energy and radiation balance in the atmosphere, including heat transfer processes, are described by rate equations, which are included in thermodynamic equations for typical conditions as found in the troposphere.
Based on the numerical result, the propagation of various wave patterns in the atmosphere might be the result of a single disturbance of an atmospheric variable.The temporal propagation of v h and w has a similar wave pattern that both rise with time at each f.The propagation of the wave associated with temperature T m along time at a mentoring point is decreasing, but has a constant value along the f direction at each time step.This result shows that increasing in both directions of wind speeds and decreasing in temperature, or vice versa, is the consequence of energy transformation throughout the system due to the dissipation function.
Following the decreasing of T m , there is a decreasing in temporal propagation that is associated with the entropy s m , pressure p m , and gas constant R m of a saturated moist air at each point of f.This implies that there is a direct relationship between the temperature, the entropy, the pressure (p m ), and the gas constant (R m ) of saturated moist air as a function of time at each point of f.In reality, the random motion of the saturated moist air is low at low temperatures, which agrees with this research result.Moreover, the reduction of p m and R m as a function of time at each f is simply the consequence of the ideal gas law (or the system is ideal gas).
The propagation associated with c pm and ρ m of the saturated moist air has a similar wave pattern; both of them are increasing as a function of time at each step of f but have a constant value along f at each time step.As we can see from equation (1), the factors that affect the value of c pm is the mixing ratios q j (j = d, v, c, r) and q j itself depends on T m of the saturated moist air as in equation (7).So, from this equation, the c pm of the saturated moist air is inversely related by T m value that clearly obtained in the result.Additionally, the inverse relation of ρ m with both T m , p m , and R m , is simply to indicate the considered system is an ideal gas that obey the ideal gas law.
The results of this study provide new insights regarding the effect of an atmospheric temperature on the dynamics of moisture at a geographical location in Ethiopia.However, the challenge of an ice phase in the atmosphere and its different conformations, such as snow, graupel, or hail, were not considered in this study.This will be further examined and analyzed in subsequent work to obtain a complete picture of the outcome.
Appendix.Some detailed aspects of the numerical model that were used in the paper are given in the discussion below A.1.Microphysics parameterization The methods of Klemp and Wilhelmson (1978) [53] are used in the current investigations for cloud parameterization, and the rates (kgm −3 s −1 ) of cloud water by autoconversion, S ac , collection, S cr , and evaporation, S ev , into rainwater is provided as where the saturation vapor density is represented by v r*, and the parameters f ice and f vent are specified as follows in [53]: The rainfall speed v rain as well as the diffusion flux  q j of the air parcel that introduced in equation ( 16) is given by: with μ T and ν T as in equation (17).Here, q j and g  are respectively the mixing ratio of the moist air and the acceleration due to gravity of the Earth.

A.2. Calculation of condensation
The rate at which cloud water condenses S cd when the air is saturated is computed as follows from [59,60]: where the rate change of the saturated mixing ratio is denoted by q t vs ¶ ¶ , and the local increases in the water vapor mixing ratio are represented by Where L H denotes the latent heat of the air parcel that vary with the temperature of moist air.
According to [59,60], the second term on the right in (A6) denotes the subgrid-scale turbulent diffusion of q v .It is as follows: Other physical constant value include κ, f, and σ are provided in table 1.Where θ o , q¢ , θ e , and θ v are the potential temperature at the base, perturbed state, the equivalent potential temperature and virtual potential temperature respectively : these are defined by with p m ,T m as in equation (2) In the computation of q vs from Clapeyron-Clausius equation in (A7), the base state pressure considered in equation (10).The term q vs in (A7) is expressed as:  and L H as in equations (2), ( 8), (10), and (A8), respectively.
A.3.Heat fluxes of the ground surface (Q  ) When we consider the interaction of the ground with the air parcel, the following heat fluxes occur from the ground surface to our system: A.3.1.Convective heat flux.According to [61], the convective heat flux is determined by the heat transfer equation: where, h is the coefficient of convective heat transfer, T a is the average daily temperatures of the air and the ground surface temperature changes in the annual cycle.Their corresponding expression are given as follows: where u is the wind velocity of the moist air, ρ m is its density and c p is the specific heat of the air as in equation (1).
Wherein the values of the constants parameters are p a = 0.270 rad and p s = 0.166 rad [58].
A.3.2.Radiation flux.As described in [62,63], the net radiative heating rate that contain both a short and longwave can be given as : The value of S m , A sol , and ω are available in table 1 and P sol = − 0.153rad.
A.3.3.Evaporative heat flux.According to [64], the heat transferred between the surface of the ground and the environment as a result of evaporation of water from the ground can be expressed as:  [46-48] and moist air temperature as in equation (2), the conductive heat flux is obtained using Fourier's law: Considering the relationships (A13), (A15), (A16), and (A17) in the heat balance equation (23), it is obtained: Here, r is the distance from the center of the Earth surface r R z e ( ) = + and for this specific studies, we have the Earth and frequency, s −1  v , rain q r  terminal velociy of the rain and the diffusin flux of the air parcels c l specific heat of liquid water at STP c pd , c pv specific heat of dry air and water vapor at STP at constant pressure in Jkg −1 K −1 c pm , c vm specific heat of saturated moist atmosphere at constant pressure and volume, in Jkg −1 K −1 , respectively c vd , c vv specific heat of dry air and water vapor at STP at constant volume in Jkg −1 K −1 f ice , f vent coefficients defined by the formula (A2) G, J n the universal gravitational and Jefferyʼs constants h, h m convective heat transfer coefficient and enthalpy of the saturated moist air k T , μ T temperature dependent ransport coeffients defined by formula (17), LH oo latent heat of the air at 0K temperaure M mass of the Earth m d , m w , ε molecular weight of dry air and water p p , vo vo ¢ follows by entering μ T into equation (21) and the result into equation (20):

Figure 5 .
Figure 5.The pressure p m propagation of the saturated moist air following disturbance at the mentoring point [λ, z] = [39.24689°,6.344 km].

Figure 6 .
Figure 6.The propagation of the saturated moist air's gas constant R m following a temperature perturbation at a mentoring point [λ, z] = [39.3°,6.344 km].

Figure 7 .
Figure 7.The specific heat capacity at a constant pressure (c pm ) propagation of saturated moist air following disturbance at a mentoring point [λ, z] = [39.3°,6.344 km].
6with h, T a and T s as in equation (A14).The term ξ is the coefficient of evaporation rate and RH is the relative humidity of ambient air.Their corresponding value as well as the constant parameters C EV , A, and B are incorporated in table1A.3.4.Conductive flux.By considering temperature dependent thermal conductivity K T m