The Effect of Quadric Shear Zonal Flows and Beta on the Downstream

In this paper, the influence of quadric shear basic Zonal flows and β on the downstream development of unstable chaotic baroclinic waves is studied from the two-layer model in wide channel controlled by quasi geostrophic potential vorticity equation. Through the obtained Lorentz equation, we focused on the influence of the quadric shear zonal flow (the second derivative of the basic zonal flow is constant) on the downstream development of baroclinic waves. In the absence of zonal shear flow, chaotic behavior along feature points would occur, and the amplitude would change rapidly from one feature to another, that is, it would change very quickly in space. When zonal shear flow is introduced, it will smooth the solution of the equation and reduce the instability, and with the increase of zonal shear flow, the instability in space will increase gradually. So the quadric shear zonal flow has great influence on the stability in space. Keyword: β effect; quadric shear zonal flow; baroclinic instability; Lorentz dynamics 10


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The downstream development of linear and nonlinear instability has a long history in hydrody-26 namics. In the actual atmosphere, the great development of general large-scale motion is often 27 related to the baroclinic nature of the atmosphere. Therefore, it is necessary to discuss the in-  (Pedlosky, 1976;Polvani and Pedlosky, 1988) have made a lot of discussions on it and obtained a 36 broad research topic. In this paper, the influence of zonal shear flow and β on the development of 37 the downstream of the slope is studied. Generally, chaotic behavior appears in the unstable baro-38 clinic system, and its performance needs to be studied in the unstable development environment. parameter domain, the time change of the system shows that it is extremely unstable to the initial 46 data, so from the perspective of time change alone, the initial data that we evolved for each feature 47 according to the Lorenz model has slightly different adjacent characteristics.When the adjacent 48 features of chaos along time and the dynamic development in the downstream coordinate system 49 are introduced into it, we will get the first-order divergent solution. Because the fast change of the 50 behavior in the downstream coordinate system is not caused by the range of the system character-51 istics developing from parallel to chaos, the impact of chaos is different from the common impact 52 in the hydrodynamics. Because in the β effect, the unstable solution at the origin of the solution 53 phase plane tends to be shielded from the trajectory, so for the small value of β , the solution is also 54 asymptotic to the periodic solution. The β parameters are regarded as a smally but importment 55 disturbance to the dynamic. Without the β effect, the two-layer model with uniform vertical shear 56 is unstable. The stronger the vertical wind shear is, the more favorable it is toproduce the baro-  [∇ 2 ψ n + F(−1) n (ψ 1 − ψ 2 )] + J[ψ n , ∇ 2 ψ n + F(−1) n (ψ 1 − ψ 2 ) + β y] = −r∇ 2 ψ n , (2.1) where n = 1, 2, the rotational Froude number can be expressed as F = f 2 L 2 /g D, f is the Cori- Where U B and U T are related to latitude y and the functions ϕ B and ϕ T are the barotropic and 76 baroclinic perturbation streamfunctions. From equations (2.1), the perturbations ϕ B , ϕ T satisfy where since the upper and lower basic zonal flow are quadric shear, d 2 U B dy 2 and d 2 U T dy 2 are constants.

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F and F c are the same as employed in Pedlosky(2019), give the critical curve of instability in the 79 form of lowest order as a relation between F c , the critical value of F , that is, where the wave number K 2 = k 2 + l 2 .

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For small values of r the minimum occurs at very long wavelengths and we need to consider the 82 scale of the problems variables. The following assumptions: The basic flow is only slightly super-critical with respect to F so that The absolute potential vorticity gradient of the layer model and dissipation are also small,

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(iii) The processes of the generated disturbance systems, such as the slowly varying trough systems 86 and cyclones after being generated in the real atmosphere and ocean, are carried on more slowing 87 than their generating processes, therefore the solution of the equations (2.3a,b) will be a function 88 of "fast" and "slow" space and time variables. In such case, using ξ to represent a new fast spatial 89 coordinate, X to represent a new slow space coordinate, τ to represent a new fast time coordinate 90 and T to represent a slow time coordinate, each defined by (2.6b) The perturbations streamfunctions ϕ B , ϕ T will expand the progressive series in the small ampli- where * denotes the complex conjugate of the preceding expression.

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At the next order in O(ε 2 ) we get an expression for the baroclinic perturbation, × e ik(ξ −cτ) sin πy + * + Φ(X, y, T ), (2.9) In (2.9), the final term Φ(X, y, T ) is the baroclinic correction to the mean flow and is a function of 99 only the slow space-time variables X and T , as well as y.

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According to the above expressions, the nonlinear interaction terms, namely the Jacobian of the 101 next order, can be calculated and obtain as the governing equation for Φ. (2.11) After the equation is modified by the baroclinic mean flow, the solvable condition of O(∆ 3/2 ) 106 can be determined by the evolution governing equation of amplitude A. After we obtain where (Pedlosky, 2019) the governing equations (2.11) and (2.12) to be rewritten (after dropping primes from the new 110 dependent variables) as We finally obtain five equations, Defining the characteristic coordinate s by the differential relations (Pedlosky, 2011(Pedlosky, , 2019 116 (2.17a-e) can be written as the set of first order ordinary differential equations 117 dA r ds =Ā r ,

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Since equation (2.14) is affected by the boundary condition X = 0, we choose as Where T period , a, γ, b will be given (Pedlosky, 2019).

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When γ is sufficiently small,the Lorenz dynamics along the characteristics of the partial dif-123 ferential equations of (2.14) produced chaotic solutions. For development problems in space and 124 time , resulting in a value of A at a given time, which changes suddenly with X.

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In Fig.1. When b = 0.4, the instability of the real part of A is relatively strong. When b increases 126 to 1.2, the instability of the real part of A gradually decreases.When b = 6, it can be seen that when 127 b is large enough, the real part of A tends to be stable, indicating that zonal shear flow enhances 128 the stability of the real part of A.

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In Fig.2. When b is small, the real part of R tends to be stable, and when R suddenly increases  When n = 1, 2 Eq.(2.1) We insert Eqs.(A.1) into Eq.(A.3) to obtain the perturbation streamfunctions ϕ B , ϕ T , respectively,