Magnetic field annihilation and charged particle acceleration in ultra-relativistic laser plasmas 5
π
i
= p
i,0
/m
e
c, respectively, and µ = m
e
/m
p
is the electron
to ion mass ratio. We assume that m
i
= m
p
.
Equation (12) can be written by using the d’Alembert
relationship, giving the solution of the Cauchy problem for
the wave equation, as
a
(
x,t
)
=
−ω
0e
Z
t−|x|/c
0
(
a
0,t
′
+π
e
p
1 +
[
a
(
0,t
′
)
+π
e
]
2
+µ
a
0,t
′
−π
i
p
1 +µ
2
[
a
(
0,t
′
)
−π
i
]
2
)
dt
′
+
2πeI
0
m
e
c
2
[|x | +θ(ct −|x|)(ct −|x|)]
−
eE
w
m
e
c
3
[
(
ct +x
)
θ
(
ct +x
)
+
(
ct −x
)
θ
(
ct −x
)
]
. (14)
To find the function a
(
0,t
)
we set x = 0 in
Equation
(14)
and differentiate it with respect to time t. The result
is an ordinary differential equation of the first order whose
solution is
Z
a
0
(
0,t
)
"
a
′
+π
e
p
1 +
(
a
′
+π
e
)
2
+µ
a
′
−π
i
p
1 +µ
2
(
a
′
−π
i
)
2
−
2πeI
0
m
e
c
2
ω
0e
−
2eE
w
m
e
c
2
ω
0e
−1
da
′
= −ω
0e
t. (15)
One may see from
Equation (15) that if the amplitude
of the electromagnetic wave is substantially large, E
w
>
4πen
0
l −π I
0
/c, there can be no steady state solution. In the
limit t → ∞ the electromagnetic potential at x = 0 grows
proportionally to time as
a
(
0,t
)
≈ −
et
m
e
c
h
E
w
−
π
c
(
4en
0
lc −I
0
)
i
. (16)
Consequently, the unlimited acceleration of charged parti-
cles occurs with the particle momentum growing as
p
z
(
0,t
)
≈ −et
h
E
w
−
π
c
(
4en
0
lc −I
0
)
i
. (17)
Substituting the function a
(
0,t
)
found from
Equation (15)
to (14) we obtain the expression describing the electromag-
netic field outside the current sheet. In the case of super-
strong electromagnetic field, when E
w
> 4π en
0
l−πI
0
/c, t he
electric field has a form of two almost constant waves with
the amplitude E
w
−4πen
0
l + π I
0
/c propagating outwards
in two directions. For the electromagnetic wave incident on
the current sheet with moderate amplitude, E
w
< 4π en
0
l −
πI
0
/c, the wave is reflected from the current sheet. In the
region | x |< ct the electr i c field vanishes and the magnetic
field is equal to B
y
=
(
B
0
+E
w
)
sgn(x). The momentum of
charged particles increases a finite value.
We note that the solution found here corresponds to the
paradigm of the thin foil nonlinear electrodynamics devel-
oped in Refs. [
89,91–94].
2.3. Tearing mode instability of thin current layer
The 2D dynamics of thin current sheet is considered within
the framework of the tearing mode instability concept. The
electromagnetic tearing mode instability of thin current sheet
has been studied in Refs. [
37,89,95–97]. The electromagnetic
field given by the vector A =
m
e
c
2
a/e
e
z
and electrostatic
potential ϕ = m
e
c
2
φ/e is governed by the equations
Wa =δ(x)J, (18)
Wϕ = δ(x)R, (19)
where W = ∂
xx
+∂
yy
−c
−2
∂
tt
is the d’Alambertian,
J = 4π
X
j=e,i
e
j
n
j
lv
z,j
/m
j
c
2
(20)
and
R = 4π
X
j=e,i
e
j
n
j
l/m
j
c
2
(21)
are the surface electric current and electric charge density,
respectively. Here it is assumed that the variables a and
ϕ depend on the coordinates x, y and time t. Lineariz-
ing Equations (
18) and (19) and presenting dependence of
the potentials on time and coordinates in the form a
(1)
=
a
(1)
(x) exp
(
−ωt +ikx
)
and ϕ
(1)
= ϕ
(1)
exp
(
−ωt +ikx
)
, we
can write the Equations (
18) and (19) as
d
2
a
(1)
dx
2
−Q
2
a
(1)
= δ(x)
J
a
a
(1)
+J
ϕ
ϕ
(1)
, (22)
d
2
ϕ
(1)
dx
2
−Q
2
ϕ
(1)
= δ(x)
R
a
a
(1)
+R
ϕ
ϕ
(1)
, (23)
with J
a
= ∂
a
J, J
ϕ
= ∂
ϕ
J, R
a
= ∂
a
R, and R
ϕ
= ∂
ϕ
R, where
Q =
p
k
2
−ω
2
/c
2
. (24)
The solution of Equations (
22) and (23) yields
a
(1)
(x) = −
exp
(
−Q|x|
)
2Q
J
a
a
(1)
(0) +J
ϕ
ϕ
(1)
(0)
, (25)
ϕ
(1)
(x) = −
exp
(
−Q|x|
)
2Q
R
a
a
(1)
(0) +R
ϕ
ϕ
(1)
(0)
. (26)
Setting x = 0 in both sides of Equations (
25) and (26) we
obtain the system of algebraic equations. The condition of
剩余20页未读，继续阅读
weixin_38729399
- 粉丝: 7
- 资源: 902
我的内容管理
展开
最新资源
- 计算机人脸表情动画技术发展综述
- 关系数据库的关键字搜索技术综述：模型、架构与未来趋势
- 迭代自适应逆滤波在语音情感识别中的应用
- 概念知识树在旅游领域智能分析中的应用
- 构建is-a层次与OWL本体集成：理论与算法
- 基于语义元的相似度计算方法研究：改进与有效性验证
- 网格梯度多密度聚类算法：去噪与高效聚类
- 网格服务工作流动态调度算法PGSWA研究
- 突发事件连锁反应网络模型与应急预警分析
- BA网络上的病毒营销与网站推广仿真研究
- 离散HSMM故障预测模型：有效提升系统状态预测
- 煤矿安全评价：信息融合与可拓理论的应用
- 多维度Petri网工作流模型MD_WFN：统一建模与应用研究
- 面向过程追踪的知识安全描述方法
- 基于收益的软件过程资源调度优化策略
- 多核环境下基于数据流Java的Web服务器优化实现提升性能
资源上传下载、课程学习等过程中有任何疑问或建议，欢迎提出宝贵意见哦~我们会及时处理！
点击此处反馈
