n=0
R0n(· )n (A.21)
Substitute this into governing equations and group terms by order of R0. The 0th order terms give
V⃗0 = ˆk× ∇hϕ0 (A.22)
∂ϕ0
∂z = b0 (A.23)
∂b0
∂t + ⃗V0· ∇hb0+ w0∂b0
∂z = Q0 (A.24)
The 1st order terms give
∂ ⃗V0
∂t + ⃗V0· ∇hV⃗0+ ˆk× ⃗V1+ βyˆk× ⃗V0+∇hϕ1 = ⃗F1 (A.25)
∂ϕ1
∂z = b1 (A.26)
∇h· ⃗V1+ ∂ρw0
ρ∂z = 0 (A.27)
∂b1
∂t + ⃗V0· ∇hb1+ ⃗V1· ∇hb0+ w0∂b0
∂z = Q1 (A.28)
The above equations form the governing equations of quasi-geostrophic thoery.
A.2 Balanced Condition
If the flow is zonally periodic and balanced in y-direction
∂v0
∂t + ⃗V0· ∇hv0 = 0 (A.29)
then take zonal mean governing equations of 0th-order
The 1st order terms give
∂u0
Collect suitable equations from above (0th order equations, zonal momentum equa-tion and continuity from 1st order equaequa-tions) we finally get
∂u0
dimension to get more conventional notations
∂ug
∂t − f0va= F −∂vg′u′g
∂y = F∗ (A.43)
∂b
∂t + w∂b
∂z = Q−∂v′b′
∂y = Q∗ (A.44)
∂va
∂y +∂ρw
ρ∂z = 0 (A.45)
∂ϕ
∂z = b (A.46)
ug =−1 f0
∂ϕ
∂y (A.47)
The equation above are the governing equations given in section 2.1.
APPENDIX B
WAVES AND THE ELIASSEN-SAWYER CIRCULATION EQUATION
Eliassen circulation equation is primarily based on balanced state, in which buoyancy waves cannot exist. In this appendix we will use perturbation method to include buoyancy wave. We will also see that the frequency product of horizontal and vertical direction is a constant constrained by Jacobian of angular momentum and buoyancy.
Consider the following primitive equations in Cartesian space which is symmetric in x-direction and its vertical coordinate z is physical height
∂u
in which sound waves are eliminated by letting dρ/dt = 0.
We define “geostrophic state” as the state when
d
dt(· ) = 0, (B.2)
where
Since t does not appear explicitly in (B.4a)-(B.4d), we set
∂
∂t(· ) = 0. (B.5)
Thermal wind relation can be derived through (B.4b) and (B.4c) as
f order less than others, then we obtain
f∂u
∂z ≈ −g θ
∂θ
∂y. (B.7)
Integrating (B.4d) with z, we obtain
w = constant = 0. (B.8)
otherwise w̸= 0 on top and bottom boundaries.
Letting all variables be perturbed by a small amplitude, i.e. (· ) = ( · ) + ( · )′, and
retaining the first order of (B.1a)-(B.1e) , we obtain
in which we approximate
ρ′
ρ ≈ −θ′
θ. (B.10)
under the physical sense that sound wave is relatively fast in our scale. Equation (B.9d) enables us to write
(ρv′, ρw′) =
After taking z and y derivative of (B.9b) and (B.9c), respectively, we derive an modified version of thermal wind relation
f∂ρu′
Static stability: A = g θ
Inertial stability: C = f (
f −∂u
∂y )
. (B.13c)
to rewrite (B.9a) and (B.9e) as
Adding ∂(B.14a)/∂z and ∂(B.14b)/∂y, substituting (B.11a) and (B.12) into it, we get
∂2
Neglecting Q, F and assuming A, C are constants, dispersion relation given by (B.15a) is
ω2(
l2+ m2)
= Al2+ 2Blm + Cm2, (B.16)
where ω, l and m are wavenumber of t, y and z. (B.16) reduces to purely dispersion relation in usual buoyancy wave system if there is no baroclinity, i.e. B = 0,
ω2(
where κ =√
l2+ m2. Substituting (B.18b)-(B.18a) into (B.16), we get
ω2 = A sin2θ + B sin θ cos θ + C cos2θ
where α is designed as
sin α = √(C− A) /2
to apply sum-to-product identities.
To get the group velocity, we first notice
∇ =
Applying (B.21) to (B.16), we get
(cgx, cgy) = ∇ω
So the direction of energy transport is orthogonal to the wave geometry whose amplitude is controlled by A, B, and C.
Possible range of ω2is
The stable solution requires ω2 > 0,
C + A
which is equivalent to require the time-independent part of (B.15a) to be elliptic. The maximum and minimum frequencies are
ωmax= C + A
and their product is
ωmax ωmin= AC − B2. (B.27)
(B.27) implies the stiffnes of the system is controlled by AC − B2 variable only. If on direction gets stiffer, it will be accompanied by the relaxation of the other direction.
The quantity AC − B2is actually the Jacobian of absolute angular momentum and buoyancy force
which is shown as follows
AC− B2 = g θ
∂θ
∂zf (
f− ∂u
∂y )
+ f∂u
∂z g θ
∂θ
∂y
= ∂b
∂z
∂m
∂y −∂m
∂z
∂b
∂y
= ∂( m, b)
∂ (y, z) . (B.29)
The condition (B.15b) is elliptic if B2 − AC can be realized through the example shown in Fig. 19, 20, and through the use of B.29. m controls the movement in y direc-tion, when an air parcel is displaced from its original posidirec-tion, it will oscillate back to its original y position. The same is also true for b, but in z direction. The composite effect gives Fig. 19 a stable configuration in which the displaced air parcel tends to go back its original position, corresponding to (B.29) being positive (elliptic), and Fig. 20 an unsta-ble configuration in which the displaced air parcel tends to move away from its original position, corresponding to (B.29) being negative (non-elliptic).
Figure 19: A stable configuration of m and b. The red and blue arrows are buoyancy restorcing force and inertial restoring force, respectively.
Figure 20: An unstable configuration of m and b. The red and blue arrows are buoyancy
APPENDIX C
BOUNDARY CONVERSION
In the derivation of efficiency, we apply boundary condition for rψ and rχ to get the de-sired self-adjoint property (equation [2.71]). However when dealing with open boundary condition, i.e. when Ekman pumping is considered, extra terms appear.
If we restrict ourselves by considering bottom-opened scenario (which is reasonable since in a tropical cyclone dynamical influence usually comes from bottom), we will re-quire rψ → 0 as r → 0, ∞ or as z → z∞. Apply integral by parts to LHS of equation (2.71)
∫∫
ψLχ rdrdz =
∫∫
χLψ rdrdz +
∫ ( Cψ∂χ
∂z ) z∞
z0
rdr (C.1)
The second term due to boundary condition is called boundary conversion, denoted as CB, characterizing the effect of boundary condition on rψ (vertical motion). This term depends only on boundary condition and χ (or temperature profile θ (r, z)) but not on diabatic heating Q nor external forcing F .
APPENDIX D
SIMILARITY BETWEEN CYLINDRICAL AND SPHERICAL COORDINATES
In this appendix we would prove that governing equations in cylindrical coordinates (Sec.
2.3) and spherical coordinates (Sec. 2.4) are essentially identical but with minor differ-ence.
For clarity we list again the governing equations in both coordinates [(2.59) and (2.83)] and arrange them in proper order. We also redefine geopotential ϕ in cylindrical coordinates as G to avoid obfuscation.
Cylindrical coordinates:
After speculation and with the aid of Fig. 21, we notice that there are two types of “radius”.
One is the coordinate radius ˜r while the other is the radius ˜R (˜r) with respect to the rotation
Figure 21: The comparison between cylindrical and spherical coordinates.
center. The following gives more general governing equations.
Absolute angular momentum: ∂m
∂t + ˜u∂m
∂ ˜r + w∂m
∂z = ˜RF , (D.3a) Thermodynamic: ∂θ
∂t + ˜u∂θ
∂ ˜r + w∂θ
∂z = Q, (D.3b)
Balanced condition:
S (˜˜ r) R˜3
(
m2− ˜Ω2R˜4 )
= ∂G
∂ ˜r, (D.3c) Hydrostatic: ∂G
∂z = θ θ0
g, (D.3d)
Continuity: ∂ ˜R˜u
R∂ ˜˜ r + ∂ρw
ρ∂z = 0. (D.3e)
where m = ˜R˜v + ˜Ω ˜R2 is the absolute angular momentum, ˜v is the main circulation,
˜
u = d˜r/dt together with w represents the secondary circulation, and ˜S (˜r) is a parameter depending only on ˜r.
By (D.3c) and (D.3d), we derive the thermal wind relation
in which we notice that the parameter ˜S is a minor issue since it depends only on horizontal direction so it would not involve in the operation when deriving thermal wind relation.
After taking time derivative of (D.4), we get
∂
Static stability: ρA = g θ0
Inertial stability: ρC = 1 R˜3
∂m2
∂ ˜r , (D.6c)
to rewrite (D.3a) and (D.3b) as
1
According to (D.3e) we define the streamfunction ψ such that
(ρ˜u, ρw) =
After subtracting ∂(D.7b)/∂ ˜r from ∂(D.7a)/∂z to eliminate partial derivative of time with the aid of (D.5) and substituting (D.8) into it, we obtain a generalized Eliassen-Sawyer circulation equation for both coordinates
Lψ = g ∂Q
− 1 ∂2mF
, (D.9a)
where
We close this part by deriving Eliassen operators for both coordinates. Using the variables defined in Table 6, we get the Eliassen operators for cylindrical coordinates
L (· ) = ∂
Static stability: ρA = g θ0
Inertial stability: ρC = 1 r3
Static stability: ρA = g θ0
Inertial stability: ρC =−sin ϕ R3
∂m2
∂(aϕ). (D.13c)
Table 6: General form of Eliassen-Sawyer circulation equation in cylindrical and spherical coordinates
Variables in (D.9b) Cylindrical Spherical
˜
r r aϕ
R˜ r R
Ω˜ f /2 Ω
˜
v v u
˜
u u v
S˜ 1 − sin ϕ
APPENDIX E
APPLICATION PROGRAMMING INTERFACE
Here we list the core API to solve (3.1). The code is written in Fortran 95 and maintained on Github (http://github.com/meteorologytoday/XLab-EE-fortran).
Basic usage is as follows
1 ! This is a simple example of solving Eliassen-Sawyer 2 ! circulation equation. All other variables are set 3 ! initially with the correct type specified by API below 4
5
6 ! #1: Calculate coefficient matrix first 7 err_flg = 0
8 call cal_coe(a, b, c, coe, dx, dy, nx, ny, err_flg) 9 if (err_flg /= 0) then ! Error occurs.
10 exit 11 end if 12
13 ! #2: Solve the equation 14 err_flg = 0
15 strategy = 0 16 strategy_r = 1e-3
17 call solve_elliptic(max_iter, strategy_r, 1.0, dat, coe, f, &
18 & workspace, nx, ny, err_flg, debug) 19 if (err_flg /= 0) then ! Error occurs
20 exit 21 end if
#Subroutine:
cal_coe (a, b, c, coe, dx, dy, nx, ny, err)
#Description:
Calculate (3.8a) and store result in coe.
#Parameters:
– Real(4) :: a (nx-1, ny-2)
Static stability A in (3.1) whose dimension is (nx-1, ny-2).
– Real(4) :: b (nx-1, ny-1)
Static stability B in (3.1) whose dimension is (nx-1, ny-1).
– Real(4) :: c (nx-2, ny-1)
Static stability C in (3.1) whose dimension is (nx-2, ny-1).
– Real(4) :: coe (9, nx, ny)
Coefficient matrix which stores the result of this subroutine. Its dimension is (9, nx, y).
– Real(4) :: dx
Grid spacing in x direction.
– Real(4) :: dy
Grid spacing in y direction.
– Integer :: nx
Number of grid points in x direction.
– Integer :: ny
Number of grid points in y direction.
– Integer :: err
Error flag. Modified to 0 if completed and without error, otherwise not 0.
#Subroutine:
do_elliptic (psi, coe, outdat, nx, ny, err)
#Description:
Calculate Lψ in (3.1). Result is stored in outdat.
#Parameters:
– Real(4) :: psi (nx, ny)
ψ field whose dimension is (nx-1, ny-2).
– Real(4) :: coe (9, nx, ny)
Coefficient matrix calculated beforehand by Subroutine cal_coe.
– Real(4) :: outdat (nx, ny) Result of Lψ in (3.1).
– Integer :: nx
Number of grid points in x direction.
– Integer :: ny
Number of grid points in y direction.
– Integer :: err
Error flag. Modified to 0 if completed and without error, otherwise not 0.
#Subroutine:
solve_elliptic(max_iter, strategy, strategy_r, alpha, dat, coe, f, workspace, nx, ny, err, debug)
#Description:
Invert ψ in (3.1). Boundary conditions are given in the boundaries of f. Result is stored in dat. This subroutine now provides two ways to judge the convergence which can be specified with strategy.
“1” specifies to judge “absolute” residue defined in (3.10) and this critical value should be given in strategy_r.
“2” specifies to judge “relative” variation of residue defined in (3.10) and this critical value should be given in strategy_r.
#Parameters:
– Integer :: max_iter
Maximum iteration time. If iteration time is reached and convergence crite-ria is not met, then err̸= 0.
– Integer :: strategy
Strategy used to judge convergence. See #Description part.
– Real(4) :: strategy_r
This value service different criteria according to strategy. See #Description part.
– Real(4) :: alpha
Over-relaxation parameter. It is recommended to set this value 1.0.
– Real(4) :: dat (nx, ny)
Result of relaxation. The initial guess of iteration can be placed in this array.
Its dimension is (nx, ny).
– Real(4) :: coe (9, nx, ny)
Coefficient matrix calculated beforehand by Subroutine cal_coe.
– Real(4) :: f (nx, ny)
This is F in (3.1) whose dimension is (nx, ny). Notice that boundary condi-tions are given in the boundaries of f.
– Real(4) :: workspace (nx, ny)
The workspace when doing relaxation whose dimension is (nx, ny).
– Integer :: nx
Number of grid points in x direction.
– Integer :: ny
Number of grid points in y direction.
– Integer :: err
Error flag. Modified to 0 if completed and without error, otherwise not 0.
– Integer :: debug
Debug message output if 1. No debug message if 0.
BIBLIOGRAPHY
Charney, J. G., and A. Eliassen, 1964: On the growth of the hurricane depression. J. Atmos.
Sci., 21, 68–75.
Hack, J. J., and W. H. Schubert, 1986: Nonlinear response of atmospheric vortices to heating by organized cumulus convection. J. Atmos. Sci., 43, 1559–1573.
Hack, J. J., W. H. Schubert, D. E. Stevens, and H.-C. Kuo, 1989: Response of the hadley circulation to convective forcing in the itcz. J. Atmos. Sci., 46, 2957–2973.
Holliday, C. R., and A. H. Thompson, 1979: Climatological characteristics of rapidly intensifying typhoons. J. Atmos. Sci., 107, 1022–1034.
Hoskins, B. J., and F. P. Bretherton, 1972: Atmospheric frontogenesis models: Mathe-matical formulation and solution. J. Atmos. Sci., 29, 11–37.
Hoskins, B. J., and N. V. West, 1979: Baroclinic waves and frontogenesis. part ii: Uniform potential vorticity jet flows-cold and warm fronts. J. Atmos. Sci., 36, 1663–1680.
Kuo, H.-C., C.-P. Chang, Y.-T. Yang, and J. H.-J., 2009: Western north pacific typhoons with concentric eyewalls. Monthly Weather Review, 137, 3758–3770.
Rozoff, C. M., W. H. S. Schubert, and J. P. Kossin, 2008: Some dynamical aspects of tropical cyclone concentric eyewalls. Quart. J. Roy. Meteor. Soc., 134, 583–593.
Schubert, W. H., P. E. Ciesielski, C. Lu, and R. H. Johnson, 1989: Dynamical adjustment of the trade wind inversion layer. J. Atmos. Sci., 52, 2941–2952.
Schubert, W. H., and J. J. Hack, 1982: Inertial stability and tropical cyclone development.
J. Atmos. Sci., 39, 1687–1697.
Schubert, W. H., and B. D. McNoldy, 2010: Application of the concepts of rossby length and rossby depth to tropical cyclone dynamics. J. Adv. Model. Earth Syst., 2, 13pp., doi:10.3894/JAMES.2010.2.7.
Schubert, W. H., C. M. Rozoff, J. L. Vigh, B. D. McNoldy, and J. P. Kossin, 2007: On the distribution of subsidence in the hurricane eye. Quart. J. Roy. Meteor. Soc., 133, 595–605.
Shea, D. J., and W. M. Gray, 1973: The hurricane’s inner core retion, i: Symmetric and asymmetric structure. J. Atmos. Sci., 30, 1544–1564.
Simpson, R. H., and L. G. Starrett, 1955: Further studies of hurricane structure by aircraft reconnaissance. Bull. Am. Meteorol. Soc., 36, 459–468.
Yang, Y.-T., E. A. Kuo, H.-C. Hendricks, and M. S. Peng, 2013: Structural and intensity changes of concentric eyewall typhoons in the western north pacific basin. Monthly Weather Review, 141, 2632–2648.